Answer:
Bonding theories predict
Step-by-step explanation:
how atoms bond together to form compounds. They predict what combinations of atoms form compounds and what combinations do not. Bonding theories explain the shapes of molecules, which in turn determine many of their physical and chemical properties
A credit card company charges a late fee of 10% on all late payments. last month, jamal was charged a late fee. the total amount paid, including the late fee, was $275. what was the original amount on the credit card before the late fee?
Answer:
265
Step-by-step explanation:
You subtract 275 minus 10
$247.5 was the original amount on the credit card
Lets simplify the problem,
Total amount after late fee = $275
Late fee deduction = 10%
Original amount = 275 - 10% of 275
= 275 - 27.5
= $247.5
So $247.5 was the original amount on the credit card before the late fee.
Percentage:
In mathematics, a percentage is a number or ratio that represents a fraction of 100. It is often denoted by the symbol "%" or simply as "percent" or "pct." For example, 35% is equivalent to the decimal 0.35, or the fraction.
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Using that income equation, after how many years will the income be $300? Round your answer correct UP TO TWO (2) decimal digits.
The number of years based on the equation is 24 years.
How to illustrate the information?The information is incomplete. An overview will be given. Let the income equation be represented as I = 60 + 10x where x represents the years taken.
Therefore i = 60 + 10x
60 + 10x = 300
10x = 300 - 60
10x = 240
x = 240/10
x = 24
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The product of two fractions is 9/3/ 5. if one of them is 9/ 3/7. find the other
The other fraction is [tex]1\frac{1}{55}[/tex].
The calculation to find out other fractionProvided that ,
One fraction = [tex]9\frac{3}{7}[/tex] = [tex]\frac{66}{7}[/tex]
The product of their fraction = [tex]9\frac{3}{5}[/tex] = [tex]\frac{48}{5}[/tex]
We have to find out the other fraction.
When one fraction and their product are given, the formula to find out another fraction = product / one fraction
So, the other fraction = [tex]\frac{48}{5}[/tex] / [tex]\frac{66}{7}[/tex]
= [tex]\frac{48}{5} * \frac{7}{66}[/tex]
= 336/330
= 56/55 [divided both the numerator & denominator by 6]
= [tex]1\frac{1}{55}[/tex]
Therefore it is concluded that the other fraction is [tex]1\frac{1}{55}[/tex] .
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URGENT WHOEVER ANSWERS CORRECTLY AND FAST GETS BRAINLIEST
What is the equation that can be used to find a percent of a number?
A .part = percent + whole
B. part = percent/whole
C. part = percent x whole
D. part = whole/ percent
Answer:
C
Step-by-step explanation:
Example
Find 55% of 200.
200 times 0.55 = 110
Answer: the answer is B
Step-by-step explanation:
if we want to find the percent of 700 people out of 4400 people, we would do 700/4400, which is equal to 0.16, or 16 percent.
What does the following expression mean? x <= y
As per inequality, it means that the value of x can be less than or equal to y.
Inequality in Mathematics
The two expressions that make up an equation or an inequality are connected in a mathematical statement. The equal sign (=) in an equation denotes the equality of the two expressions. The symbols >, <, ≤, or ≥ are used to denote an inequality, when the two expressions aren't always equal.
Type of Inequality
The given expression, x <= y denotes the type of inequality where the possible values of x are either less than y or equal to less. The value of x cannot be greater than y in any case.
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If the farmer isn't present, the fox cannot be left with either the dog or the goose, or both. if need be, the goose can be left with the grain provided the dog is present because the dog will guard the grain and won't eat the goose. help the farmer cross the river.
Answer:
See below
Step-by-step explanation:
Take fox across
go back to get goose
take goose across
take fox BACK
grab dog.....take dog across
go back get grain and take it across
go back get fox and take it across
Write the expanded expression in expanded form using the definition of an exponent.
2wz^2?
The expanded form of the given expression is 2 × w × z × z
Writing an expressionFrom the question, we are to determine the expanded form of the given expression
The given expression is
2wz²
To write an expression in an expanded form, we will write the expression as a repeated multiplication in the simplest way possible.
To do this, we will first expand the index in the given expression.
The index in the given expression is z²
Thus, we get
z² = z × z
Now, we will include the remaining individual terms in the expression and write them as product.
That is,
2wz² = 2 × w × z × z
Hence, the expanded form of the given expression is 2 × w × z × z
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Find the surface area of the composite figure. Round your answer to the nearest tenth if necessary.
If the length,breadth, height of cuboid are 16 m, 5 m,4 m , slant height of pyramid is 10m, height be 6 m then the surface area of composite figure be 444 [tex]m^{2}[/tex].
Given that the length,breadth,height of cuboid are 16m,5 m, 4m and slant height be 10 m and height be 6 m.
Surface area is basically sum of area of all the sides of a figure.
Surface area of composite figure = area of cuboid less area of common side + area of 2 slant sides of pyramid+area of two triangles.
=2(lb+bh+hl)-(lb)+2(slant height*breadth of cuboid)+2[1/2* length of cuboid* height of triangle)
=2(16*5+5*4+4*16)-16*5+2(10*5)+2[1/2*16*6]
=2(80+20+64)-80+2*50+2(48)
=2(164)-80+100+96
=328-80+196
=248+196
=444[tex]m^{2}[/tex]
Hence the surface area of composite figure having a cuboid and pyramid on top is 444[tex]m^{2}[/tex].
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The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum
Answer:
Sum is
Step-by-step explanation:
A sample of 148 college students at a large university reports getting an average of 6.85 hours of sleep last night with a standard deviation of 2.12 hours.
a.Verify that it is reasonable to use the t-distribution to construct a confidence interval for the average amount of sleep students at this university got last night.
b. Construct a 98% confidence interval for the average amount of sleep students at this university got last night. Use two decimal places in your margin of error.
c.. Provide an interpretation of your interval in the context of this data situation.
d.. Suppose you want to conduct a similar study at your university. Assuming that the standard deviation of this sample is a reasonable estimate of the standard deviation of sleep time at your university, how many students do you need to survey to estimate the mean sleep time of students at your university with 95% confidence and a margin of error of 0.5 hours?
The solution for the questions is mathematically given as
a)
t-distribution.
b)
the confidence interval for the mean, based on 98 percent of the sample, is ( 6.3952, 7.3048 )
c) the value of the [tex]\mu_0[/tex] is within the range of the 98 percent confidence interval for the mean, which is between 6.3952 and 7.3048, then accept H_0; otherwise, reject H _0.
d)
you should conduct a poll with around 123 students to determine the average amount of time that students spend sleeping at your institution.
What is the distribution to use?Generally, the equation for is mathematically given as
a.
In this case, the standard deviation of the population is unknown.
As a result, we make use of the t-distribution.
b)
We wish to generate a confidence interval with a 98 percent likelihood for the mean.
Because of this,
[tex](\bar{X}-t_{n-1,\alpha/2}\frac{s}{\sqrt{n}},\bar{X}+t_{n-1,\alpha/2}\frac{s}{\sqrt{n}})[/tex]
[tex](6.85-t_{148-1,0.02/2}\frac{2.12}{\sqrt{148}},6.85+t_{148-1,0.02/2}\frac{2.12}{\sqrt{148}})[/tex]
[tex](6.85-t_{147,0.01}\frac{2.12}{12.1655},6.85+t_{147,0.01}\frac{2.12}{12.1655})[/tex]
(6.3952,7.3048)
Therefore, the confidence interval for the mean, based on 98 percent of the sample, is ( 6.3952 , 7.3048 )
c )
If the value of the mu _0 is within the range of the 98 percent confidence interval for the mean, which is between 6.3952 and 7.3048, then accept H_o; otherwise, reject H_0.
d . Here, we want to determine the sample size
Therefore,
[tex]n=t_{n-1,\alpha/2}^2\frac{s^2}{E^2}[/tex]
[tex]n=t_{148-1,0.05/2}^2\frac{2.12^2}{0.5^2}[/tex]
[tex]n=t_{147,0.025}^2\frac{2.12^2}{0.5^2}[/tex]
[tex]n=2.6097^2\frac{2.12^2}{0.5^2}[/tex]
n=122.4364
In conclusion, you should conduct a poll with around 123 students to determine the average amount of time that students spend sleeping at your institution.
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A margin of sampling _______ is a statistical calculation for how accurate a poll's results are. quizlet
A margin of sampling "error" is a statistical calculation for how accurate a poll's results are.
What is sampling error?A population parameter's sampling error is the discrepancy between the sample statistic used to estimate it and the parameter itself. For instance, sampling error is the variance between a population mean and a sample mean.
The sampling errors can be placed into four categories:
population-specific error: The researcher makes a population-specific error when they are unsure of the appropriate population to sample. When respondents choose to participate in the study on their own, a selection error arises. (This limits responses to those who are interested, which spoils the results.)selection error: The sampling error for a sample chosen using a non-probability method is known as selection error. Selection error can result when participants opt to self-participate in a study and only those who are interested reply, as there may already be an underlying bias.sample frame error: When the incorrect sub-population is utilized to select a sample, a sample frame error happens. Last but not least, a non-response problem happens when potential responders cannot be reached or refuse to respond.non-response error: Non-response error is a mistake that happens when a survey doesn't receive a response to one or more questions.To know more about the sampling error, here
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Geometry: Write a formal proof, ASAP!!!
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
[tex] \sf \: \angle1 \cong \angle3[/tex][tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]
[tex]\qquad❖ \: \sf \: \angle1 \cong \angle2[/tex]
( by corresponding angle pair )
[tex]\qquad❖ \: \sf \: \angle2 \cong \angle3[/tex]
( given in the question )
[tex] \qquad \large \sf {Conclusion} : [/tex]
[tex]\qquad❖ \: \sf \: \angle1 \cong \angle3[/tex]
Identify which of the following is not equivalent to 2 3/4.
The figure that is not equivalent to 2¾ is 2¼ × 2½. That is option B.
Simplification of equivalent fractionsSimplification is a process that is used to make a fraction appear in its simplest forms.
The fraction given is = 2¾
The simplification of the fraction= 2^3/4 = 2
But simplification of 2¼ × 2½ is equal to;
2¼= 0.5
2½= 1
Therefore 2¾ is not equivalent to 2¼ × 2½ when simplified.
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You have one type of chocolate that sells for $1.50/lb and another type of chocolate that sells for $6.50/lb. You would like to have mix the $1.50/lb type with 10 lbs of the $6.50/lb type to make a chocolate mixture that sells for $2.50/lb. How much of the $1.50/lb type will you need to obtain the desired mixture
14 of the $1.50/lb type will you need to obtain the desired mixture.
According to the statement
Chocolate A that sells for $1.50/lb
Chocolate B that sells for $6.50/lb
Then in equation form the quantity of mixture will
A + B = X
then
$1.50A + $6.50(10) = X
1.50A + 65 = X -(1)
after solving the equation the value of a and b will become
A = 14.
So, 14 of the $1.50/lb type will you need to obtain the desired mixture.
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1. Which of the statements is NOT true about the illustration?
Angle DME is another name for angle 3.
MDis congruent to AE.
Angle 3 is congruent to angle 5.
MEis parallel to AD.
Answer: Option (1)
Step-by-step explanation:
[tex]\angle DMA[/tex] is another name for angle 3, but not [tex]\angle DME[/tex].
Quincy uses the quadratic formula to solve for the values of x in a quadratic equation. He finds the solution, in simplest radical form, to be x
Since the value of the discriminant (-19) < 0, no real solution(s)/root(s) exist for the equation. Thus, we can choose the first option:
"Zero, because the discriminant is negative".
A quadratic equation is a polynomial of degree 2, in a single variable x.
The standard form of a quadratic equation is ax² + bx + c = 0.
The quadratic formula is used to find the solution(s)/root(s) of this equation.
The quadratic formula is:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex]
In this formula, [tex]b^2-4ac[/tex] is called the discriminant (D).
The solution(s)/root(s) of the equation, depends on this discriminant value as follows:
When D > 0, the roots of the equation are real and distinct.When D = 0, the roots of the equation are real and equal.When D < 0, then no real roots exist.In the question, we are given that the simplest form of Quincy's equation in the radical form was,
[tex]x = \frac{-3 \pm \sqrt{-19} }{2}[/tex].
Comparing this to the quadratic formula,
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex]
we get the discriminant (D) = -19.
Since the value of the discriminant (-19) < 0, no real solution(s)/root(s) exist for the equation. Thus, we can choose the first option:
"Zero, because the discriminant is negative".
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For complete question, refer to the attachment.
32. Which of the following would be in the solution set to the system of inequalities given below?
y< 2x - 4
2-y-3
(0,-5)
Both (0,-5) and (0, -2) are solutions
(0,-2)
No solution
Answer:
no
Step-by-step explanation:
The two non-parallel sides of an isosceles trapezoid are each 7 feet long. The longer of the two bases measures 22 feet long. The sum of the base angles is 140°.
A) Use the Law of Cosines to find the length of the diagonal.
A) Use the Law of Sines to find the length of the shorter base.
Round your answers to the nearest hundredth.
You must show all of your work to receive credit.
The length of the diagonal and the shorter base are 17. 24 feet and 18. 7 feet long respectively.
How to determine the lengthThe cosine rule is given as;
[tex]c = \sqrt{a^2 + b^2 - 2ab cos \alpha }[/tex]
c = length of the diagonal
a = base length = 22 feet
b = 7 feet
[tex]c = \sqrt{7^2 + 22^2 - 2 * 7 * 22 cos 40}[/tex]
[tex]c = \sqrt{533 - 308 * 0. 7660}[/tex]
[tex]c = \sqrt{533 - 235. 93}[/tex]
[tex]c = \sqrt{297. 072}[/tex]
[tex]c = 17. 24[/tex] feet
Using sine rule
[tex]\frac{a}{sin A } = \frac{c}{sin C}[/tex]
[tex]\frac{a}{sin 70} = \frac{17. 24}{sin 40}[/tex]
Cross multiply
[tex]a[/tex] × [tex]sin 60[/tex] = [tex]c[/tex] × [tex]sin 70[/tex]
[tex]a[/tex] × [tex]0. 8660[/tex] = [tex]17. 24[/tex] × [tex]0. 9397[/tex]
[tex]a = \frac{16. 20}{0. 8660}[/tex]
a = 18. 7 feet long
Thus, the length of the diagonal and the shorter base are 17. 24 feet and 18. 7 feet long respectively.
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TIME REMAINING
43:25
Triangles X Y Z and X prime Y prime Z prime are shown.
ΔXYZ was reflected over a vertical line, then dilated by a scale factor of One-half, resulting in ΔX'Y'Z'. Which must be true of the two triangles? Select three options.
△XYZ ~ △X'Y'Z'
AngleXZY ≅ AngleY'Z'X'
YX ≅ Y'X'
XZ = 2X'Z'
mAngleYXZ = 2mAngleY'X'Z'
Answer:
△XYZ ~ △X'Y'Z'AngleXZY ≅ AngleY'Z'X'XZ = 2X'Z'Step-by-step explanation:
Reflection over any line is a rigid transformation that does not change angles or lengths. Dilation is a transformation that changes lengths by the same scale factor, but has no effect on angles. Dilated figures are similar to the original.
ApplicationWhen ΔXYZ is reflected over a vertical line, no lengths or angles are changed. When the reflected triangle is dilated by half, we get ...
Y'X' = (1/2)YX . . . . . . . makes option 3 false
X'Z' = (1/2)XZ . . . . . . . makes option 4 true
Since no angles are changed, we have ...
Angle XZY ≅ Angle Y'Z'X' (another name for angle X'Z'Y')
... makes option 2 true
Of course, the resulting triangle is similar to the original, so ...
△XYZ ~ △X'Y'Z' . . . . . option 1 is true
give the volume of the triangular prism shown
A-440in^3
B-528in^3
C-880in^3
D-264in^3
Answer:
D. 264 in³
Step-by-step explanation:
The missing edge dimension can be found using the Pythagorean theorem. Then the volume can be found using the appropriate formula.
Edge dimensionThe missing dimension and the two given for the top triangle will satisfy the Pythagorean theorem. If the missing dimension is represented by 'a', then we have ...
a² +8² = 10²
a = √(100 -64) = 6 . . . . inches
VolumeThe area of the top triangle is ...
A = 1/2bh . . . . where b and h are the base and height dimensions
A = 1/2(6 in)(8 in) = 24 in²
The volume of the triangular prism is ...
A = Bh . . . . . where B is the area of the triangular base, and h is the prism height
A = (24 in²)(11 in) = 264 in³
The volume of the triangular prism shown is 264 in³.
How can you use the converse of the Pythagorean Theorem
to tell if a triangle is a right triangle?
Answer:
Yes. If the squares of the 2 short sides add up to the square of the hypotenuse, the triangle contains a right angle. Hope this helped.
Write a paragraph summarizing the data set , 12,13,14,15,16,17,18,19,20,21,22,23,24,25. Included unit measure,the number of data , the shape of the distribution, value of appropriate measure of center , value of appropriate measure of spread
The number of observation given in the following data set 14. This is also called the size of the population.
What is the description for the above data?It is important to note that the above data possess the minimum value of 12 and the maximum value of 25. As indicated in the attached image, the data set is normally distributed.
The measures of central tendency are given as follows;
Median = 18.5; Mean = 18.5Standard Deviation = 4.03112887Variance = 16.25Mid Range = 18.5The quartiles are given as follow:Q1 --> 15
Q2 --> 18.5
Q3 --> 22
IQR = 7
Mean Absolute Deviation (MAD) = 3.5
Min = 12
Max = 25.
It is to be noted that the above indictors are called Descriptive Statistics.
What is the purpose of descriptive statistics?
The features of a sample or data set, such as a variable's mean, standard deviation, or frequency, are described or referred to as descriptive statistics.
We can better grasp a data sample's constituent parts by using inferential statistics.
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What is a Turned triangle?
Please tell me QUICKLY.
like the place where you are applying TURNED TRIANGLE.
coz literrally it means that if the trianlge has for example angle a on its right side then we turn the triangle as a whole and make the turn such that angle a is anywhere else but not on our right side....
I believe the explanation is clear, if not do contact me
Rewrite the ratio 15: 39 as an equivalent ratio of the form 1 : n.
Give any decimals in your answer to 1 d.p.
Answer:
n = 2.6
Step-by-step explanation:
Hello!
We can create an equation with equivalent fractions. Remember that a ratio can also be represented as a fraction.
[tex]a:b = \frac ab[/tex]We can create two fractions that are equivalent to each other given the ratios.
[tex]\frac{15}{39} = \frac 1n[/tex]Solve for n[tex]\frac{15}{39} = \frac 1n[/tex][tex]\frac{5}{13} = \frac 1n[/tex][tex]5n = 13[/tex][tex]n = \frac{13}{5}[/tex][tex]n = 2.6[/tex]The value of n is 2.6.
The art club had an election to select a president nine out of the 12 members of the our club voted in the election what percentage of the members voted
Answer:
75%
Step-by-step explanation:
9 divided by 12 is 0.75, so 9 is 75 percent of 12.
I hope this helps, have a good day!
Determine if each of the following functions from {a,b, c,d} to itself is one-to-one and/or onto. Check ALL correct answers. (a) f(a)
The answer is . one-to-one.
What is the domain of a set?The domain is the set of all first elements of ordered pairs (x-coordinates). The range is the set of all second elements of ordered pairs (y-coordinates). Only the elements "used" by the relation or function constitute the range. Domain: all x-values that are to be used (independent values).
(a) The function is onto because a, b, c, and d are members of its codomain. All points are converted into various points, making it a one-to-one relationship. The response is that it is one-to-one and onto ( A and C).
(b) It is onto because a, b, c, and d are members of the function's codomain. All points are converted into various points, making it a one-to-one relationship. The response is that it is one-to-one and onto (A and B).
(C) The element b is not onto since it is not a part of the function's codomain. All points are converted into various points, making it a one-to-one relationship. One-to-one is the right response.
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The complete question is -
Determine if each of the following functions from {a,b,c,d} to itself is one-to-one and/or onto.
Check ALL correct answers.
(a) f(a)=d,f(b)=a,f(c)=c,f(d)=b
A. onto.
B. neither one-to-one nor onto.
C. one-to-one.
f(a)=b,f(b)=a,f(c)=c,f(d)=d
A. one-to-one.
B. onto.
C. neither one-to-one nor onto.
f(a)=c,f(b)=d,f(c)=a
A. one-to-one.
B. onto.
C. neither one-to-one nor onto.
Find the value of x in 1:x=6 1/4
Answer:
x = [tex]\frac{4}{25}[/tex]
Step-by-step explanation:
1 : x = 6 [tex]\frac{1}{4}[/tex] ← change to improper fraction
1 : x = [tex]\frac{25}{4}[/tex] , then
[tex]\frac{1}{x}[/tex] = [tex]\frac{25}{4}[/tex] ( cross- multiply )
25x = 4 ( divide both sides by 25 )
x = [tex]\frac{4}{25}[/tex]
Jina needs to solve a problem that involves using a different type of units for angle measure. The problem given to her to calculate 6/5 * 240
The value of the expression 6/5 * 240 is 288
How to calculate the expression?The expression is given as:
6/5 * 240
Divide 240 by 5
6 * 48
Multiply 6 and 48
288
Hence, the value of the expression 6/5 * 240 is 288
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A psychologist treats 16 patients and records the number of sessions required to complete a behavioral therapy treatment for each patient. she computes ss = 800. assuming the 16 patients constitute all patients under her care (so the population of her patients), what is the standard deviation for these data?
Hence, the standard deviation for these data is 7.1 sessions
In statistics, the standard deviation is a measure of the volatility or variance of a set of values. [1] A low standard deviation indicates that the values tend to be close to the mean of the set (also known as the mean), and a high standard deviation indicates that the values are more widely distributed. indicate.
Standard deviation can be abbreviated as SD, and in mathematical texts and equations, it is most commonly represented by the Greek lower σ (sigma) for population standard deviation or the Latin letter s for sample standard deviation. It is a target.
The standard deviation of a random variable, sample, population, dataset, or probability distribution is the square root of its variance. It's algebraically simple, but it's actually less robust than mean absolute deviation. A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data.
Expression
\ sigma = {\ sqrt {\ frac {\ sum (x_ {i}-{\ mu}) ^ {2}} {N}}}
\ sigma = Population standard deviation
N = size mother Population
x_i = Arbitrary value from population
\ mu = Population mean
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Solve algebraically:
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{Equation a.}[/tex]
[tex]\rm{2x^2 - 162 = 0}[/tex]
[tex]\huge\textbf{Solving for:}[/tex]
[tex]\rm{2x^2 - 162 = 0}[/tex]
[tex]\huge\textbf{Add \boxed{\bf 162} to both sides:}[/tex]
[tex]\rm{2x^2 - 162 + 162 = 0 + 162}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\rm{2x^2 = 0 + 162}[/tex]
[tex]\rm{2x^2 = 162}[/tex]
[tex]\huge\textbf{Divide \boxed{\bf 2} to both sides:}[/tex]
[tex]\rm{\dfrac{2x^2}{2} = \dfrac{162}{2}}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\rm{x^2 = \dfrac{182}{2}}[/tex]
[tex]\rm{x^2 = 81}[/tex]
[tex]\huge\textbf{Take the square root of \boxed{\bf 81}}[/tex]
[tex]\rm{x = \pm \sqrt{81}}[/tex]
[tex]\rm{x = 9\ or\ x = -9}[/tex]
[tex]\huge\textbf{Therefore, your answer should be:}[/tex]
[tex]\huge\boxed{\textsf{x = } \frak{9\ or } \ \textsf{x = }\frak{-9}}\huge\checkmark[/tex]
[tex]\huge\textbf{Equation b.}[/tex]
[tex]\rm{-\dfrac{1}{2}(x - 3)^2 = -2}[/tex]
[tex]\huge\textbf{Solving for:}[/tex]
[tex]\rm{-\dfrac{1}{2}(x - 3)^2 = -2}[/tex]
[tex]\huge\textbf{Simplify both sides of your equation:}[/tex]
[tex]\rm{-\dfrac{1}{2}x^2 + 3x - \dfrac{9}{2} = -2}[/tex]
[tex]\huge\textbf{Subtract \boxed{\bf -2} to both sides:}[/tex]
[tex]\rm{-\dfrac{1}{2}x^2 + 3x - \dfrac{9}{2} - (-2) = -2 - (-2)}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\rm{- \dfrac{1}{2}x^2 + 3x - \dfrac{5}{2} = 0}[/tex]
[tex]\huge\textbf{Now, we can convert the equation to:}[/tex]
[tex]\rm{-0.5x^2 + 3x - 2.5 = 0}[/tex]
[tex]\large\text{When we changed the equation entirely, we made it easier to solve}\uparrow[/tex]
[tex]\huge\textbf{Use the quadratic formula to solve.}[/tex]
[tex]\large\textsf{The quadratic formula:}[/tex]
[tex]\mathsf{x= \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}}[/tex]
[tex]\huge\textbf{Here are your labels:}[/tex]
[tex]\text{a = }\rm{-0.5}\\\\\text{b = }\rm{ 3}\\\\\rm{c = }\rm{\ -2.5}[/tex]
[tex]\huge\textbf{Your new equation:}[/tex]
[tex]\rm{x = \dfrac{-(3) \pm \sqrt{3^2 - 4(-0.5)(-2.5)}}{2(-0.5)}}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\rm{x = \dfrac{-3 \pm \sqrt{4}}{-1}}[/tex]
[tex]\rm{x = 1\ or \ x = 5}[/tex]
[tex]\huge\textbf{Therefore, your answer should be: }[/tex]
[tex]\huge\boxed{\textsf{x = }\frak{1}\ \textsf{or x = }\frak{5}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex][tex]a)\ x_1=-9,\ x_2=9\\\\b)\ x_1=1,\ x_2=5[/tex]
Step-by-step explanation:Given equations:
[tex]a)\ 2x^2 - 162 = 0\\\\b) -\dfrac{1}{2}(x-3)^2=-2[/tex]
A) 2x² - 162 = 0
Step 1: Divide both sides by 2.
[tex]\\\implies \dfrac{2x^2 - 162}{2} = \dfrac{0}{2}\\\\\implies x^2-81=0\\\\\implies x^2=81[/tex]Step 2: Take the square root of both sides (using both the positive and negative roots).
[tex]\\\implies \sqrt{x^2}=\sqrt{81}\\\\\implies x=\pm\ 9[/tex]
Step 3: Separate into two cases.
[tex]\implies x_1 = -9,\ x_2 =-9[/tex]
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B) -1/2(x - 3)² = -2
Step 1: Multiply both sides by -2.
[tex]\\\implies -2\left(-\dfrac{1}{2}(x-3)^2\right)=-2(-2)\\\\\implies (x-3)^2=4[/tex]
Step 2: Take the square root of both sides (using both the positive and negative roots).
[tex]\\\implies\sqrt{(x-3)^2}=\sqrt{4}\\\\\implies x-3=\pm\ 2[/tex]
Step 3: Separate into two cases and solve each one.[tex]1)\ x-3=-2\implies x=-2+3\implies \boxed{x=1}\\\\2)\ x-3=2\implies x=2+3\implies \boxed{x=5}[/tex]