The determinant of the given matrix in terms of the variable a is a^2 + 5a + 2.
To compute the determinant of the given matrix, we can use the Laplace expansion along the first row. Let's denote the matrix as A:
A = [1 2 -2; 0 a -1; 2 -1 a]
Expanding along the first row, we have:
det(A) = 1 * det(A11) - 2 * det(A12) + (-2) * det(A13)
where det(Aij) represents the determinant of the matrix obtained by removing the i-th row and j-th column from A.
Now let's calculate the determinant of each submatrix:
det(A11) = det([a -1; -1 a]) = a^2 - (-1)(-1) = a^2 + 1
det(A12) = det([0 -1; 2 a]) = (0)(a) - (-1)(2) = 2
det(A13) = det([0 a; 2 -1]) = (0)(-1) - (a)(2) = -2a
Substituting these determinants back into the Laplace expansion formula:
det(A) = 1 * (a^2 + 1) - 2 * 2 + (-2) * (-2a)
= a^2 + 1 - 4 + 4a
= a^2 + 4a - 3
Simplifying further, we obtain:
det(A) = a^2 + 4a - 3
= a^2 + 5a + 2
Therefore, the determinant of the given matrix in terms of the variable a is a^2 + 5a + 2
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helpppp asap !!!!!!!!!!!
The numeric value for the logarithmic expressions is given as follows:
1. [tex]\log_{6}{\left(\frac{5}{8}\right)} = -0.263[/tex]
2. [tex]\log_6{40} = 2.059[/tex]
3. [tex]\log_6{64} = 2[/tex]
4. [tex]\log_6{125} = 2.694[/tex]
a. [tex]\log_{5}{\left(\frac{11}{30}\right)} = -0.623[/tex]
b. [tex]\log_5{330} = 3.603[/tex]
c. [tex]\log_5{121} = 2.98[/tex]
How to obtain the numeric value for the logarithmic expressions?For the first expression, we have that the logarithm of the quotient is the subtraction of the logarithms, hence:
[tex]\log_{6}{\left(\frac{5}{8}\right)} = \log_6{5} - \log_6{8} = 0.898 - 1.161 = -0.263[/tex]
The same rule is applied for item a, hence:
[tex]\log_{5}{\left(\frac{11}{30}\right)} = \log_5{11} - \log_5{30} = 1.49 - 2.113 = -0.623[/tex]
For the second expression, we apply the logarithm of the product, which is the sum of the logarithms, hence:
[tex]\log_6{40} = \log_6{5 \times 8} = \log_6{5} + \log_6{8} = 0.898 + 1.161 = 2.059[/tex]
For item b, we have that:
[tex]\log_5{330} = \log_5{11 \times 30} = \log_5{11} + \log_5{30} = 1.49 + 2.113 = 3.603[/tex]
For the third expression, we have that the exponent is equals to the base, hence we just take the lower term, as follows:
[tex]\log_6{64} = \log_6{2^6} = 2[/tex]
For the fourth term, we have the rule for the logarithm of an exponent, hence:
[tex]\log_6{125} = \log_6{5^3} = 3\log_6{5} = 3 \times 0.898 = 2.694[/tex]
The same rule is applied for item c, as follows:
[tex]\log_5{121} = \log_5{11^2} = 2\log_5{11} = 2 \times 1.49 = 2.98[/tex]
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The triangles below are similar. Identify the
appropriate similarity theorem that justifies the
similarity. Justify your reasoning.
AAA(Angle angle angle) Theorem can be used to prove given triangles as similar.
As we know we have to show all three angles equally.
so we have to show the similarities between T(DNM) and T(DFE) ....
(T Means Triangle.)
(A Means Angle)
As D is a common vertex in both the triangles
SO A(NDM) = A(FDE)................Angle D .......... 1A(DNM) = A(DFE) ------- by Alternative property exterior angles are equal...........2A(NMD)= A(FED) --------- BY Alternative property interior angles are equal .........3by equations 1, 2, and 3 we can conclude that all three angles of T(DFE) = T(DNM)
SO By the AAA rule, we can say that the two Triangles are similar.
Now we can say all the sides of T(DFE) will be in all propositions of T(DNM).
Like DN/DF=DM/DE=NM/FE
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I'm actually really stuck will anyone help
Answer:
Step-by-step explanation:
104 units
Jenna has 34 pennies. Mandy has 48 pennies. Mandy says that 72 pennies in all. what should Mandy do to fix her mistake
2 3/4 is between the whole number_____and_____
Answer: 2 and 3
Step-by-step explanation:
What is the formula of triangle and square?
The formula for a triangle is the area of a triangle = ½ × base × height. The formula for a square is the area of a square = side2.
This formula is used to calculate the area of any triangle when the base and height are known. For example, if the base of a triangle is 10 cm and the height is 5 cm, the area of the triangle is 25 cm2.
The formula for a square is the area of a square = side2. This formula is used to calculate the area of any square when the side length is known. For example, if the side of a square is 10 cm, the area of the square is 100 cm2. To calculate this, we can use the formula by squaring 10 (102 = 100).
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I need help with 27 and 28 does anyone know??
Answer:
[tex]27)x {}^{3} + 3x {}^{2} - 10x - 24 \\ 28)x {}^{3} - 9 {}^{2} + 8x + 60[/tex]
Step-by-step explanation:
Greetings!!
[tex]27)(x - 3)(x + 2)(x + 4)[/tex]
Expand; (x-3)(x+2): x^2-x-6
Apply FOIL method: (a+b) (c+d) = ac+ad+bc+bd
[tex](x - 3)(x + 2) = x.x + x.2 - 3.x - 3.2[/tex]
simplify
[tex]x {}^{2} - x - 6[/tex]
[tex]x {}^{2} - x - 6.(x + 4)[/tex]
distribute parentheses
[tex] = x {}^{2} .x + x {}^{2} .4 - x.x - x.4 - 6.x - 6.4 \\ = x {}^{3} + 4x {}^{2} - x {}^{2} - 6x - 24[/tex]
simplify/ Add similar elements
[tex]x {}^{3} + 3x {}^{2} - 10x - 24[/tex]
Thus, use the same format for question number 28 and you will get the value
[tex]x {}^{3} - 9x {}^{2} + 8x + 60[/tex]
If you have any questions or there is sth unclear tag me on comment box
Hope it helps!!
2y - 2 = 4x
y = -x + 4
i need the solution please help only have a day to answer it
Answer: x=1,y=3
Step-by-step explanation:
Given,
y - 1 =2x
y = 2x + 1-----(i)
Again,
We have,
y = 2x + 1
Now replacing the value of y from (i)
2x + 1= -x +
or, 2x + x= 3
or,3x= 3
So x=1
Now,
y=2x +1
or, y=2×1+1
so y=
Answer:
x=1, y=3. (1, 3).
Step-by-step explanation:
2y-2=4x
y=-x+4
----------
2y-2=4x implies 2(y-1)=2(2x), so y-1=2x.
(-x+4)-1=2x
-x+4-1=2x
-x+3=2x
3=2x-(-x)
3=2x+x
3=3x
x=3/3
x=1
y=-1+4=3
the admision to a local fair is $10 each for adult and $6 for each child. Each ride costs $1.50 for an adult and $1 for a child.
An adult and child will each go on 7 ride. How much more did the adult spend?
Answer:
7.5
Step-by-step explanation:
10 + ( 7 × 1.5) = 20.5
6 + ( 7 × 13 ) = 13
20.5 - 13 = 7.5
Help please, I don't understand any of this
The graph solution is in the attached images, for all the systems of inequalities.
What is the graph of a function?
The graph of a function is a visual representation of the relationship between the input values (x-coordinates) and the output values (y-coordinates) of the function. In other words, it is a visual representation of the function's rule.
The graph of a function can take many different forms, depending on the function's rule. Linear functions, for example, will produce a straight line on the graph.
we have the following inequalities:
i)
y > 3x - 17,
y ≤ -4x +11.
ii)
4x - 4y > -40,
-5x + 5y < 10.
iii)
x + 3y ≥ 31,
4x - 2y ≥ -2,.
iv)
5x + y < 3,
5x + y > 37
Hence, the graph solution is in the attached images, for all the systems of inequalities.
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Triangle A”B”C” is formed using the translation (x+2, y+0) and the dilation by a scale factor of 1/2 from the origin. Which equation explains the relationship between line AB and line A”B”?
Point A: (-3,3)
Point B: (1,-3)
Point C: (-3,-3)
Answer options:
A) Line A”B” equals Line AB divided by two
B) Line AB equals Line A”B” divided by two
C) Line AB over Line A”B” equals 1/2
D) Line A”B” over Line AB equals two
Triangle ABC is dilated by a scale factor of 1/2, the line A”B” equals Line AB divided by two
What is an equation?An equation is an expression composed of variables and numbers linked together by mathematical operations.
Translation is the movement of a point up, down, left or right in the coordinate plane.
Dilation is the increase or decrease in the size of a figure.
Triangle A”B”C” is formed using the translation (x+2, y+0) and the dilation by a scale factor of 1/2 from the origin.
Since there is a dilation by a scale factor of 1/2, hence:
A"B" = AB * 1/2
A"B" = AB/2
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What does the fundamental theorem of algebra state about the equation 2x2+8x+14=0 ?
Answer: The fundamental theorem of algebra states that the equation 2x^2+8x+14=0 has at least one complex root, which means that there are numbers that can be plugged into x that make the equation equal to zero. ✅
PLEASEEEE GIVE BRANLIEST
Step-by-step explanation:
The fundamental theorem of algebra states that for any non-constant polynomial equation with real or complex coefficients, there is at least one complex root (or solution).
In the case of the equation 2x^2+8x+14=0, the theorem guarantees that this equation has at least one complex root, which means that there are numbers that can be plugged into x that will make the equation equal to zero.
So, the fundamental theorem of algebra guarantees that for the equation 2x^2+8x+14=0, there exists at least one complex solution that makes the equation true. In this case, the solutions are x = -1+i and x = -1-i ✅
THE FIRST PERSON TO ANSWER THIS QUESTION CORRECTLY NO NEED TO EXPLAIN GETS BRAINIEST!!(REALLY EASY)
write the equation of a horizontal line that goes through the point (3,7). Explain how you got your answer. I will mark you brainliest!
Answer:
y = 7
Step-by-step explanation:
The equation of a horizontal line is worth memorizing. It is:
y = anumber
(and just in case for future reference the equation of a vertical line is
x= anumber)
Well, you are given one point (3,7).That is in the form (x,y). So x is 3 and y is 7.
y is 7
y = 7
(if it helps...horizontal lines have 0 slope. So m = 0. The equation in point-intercept form is
y = mx + b
y = 0x + b
y = b this is the y= anumber) originally mentioned here)
Evaluate the following sin6A-cos6A
On solving the provided question we can say that the value of trigonometry value is - [tex](sin^2x - cos^2x )( 1- sin^2xcos^x)\\[/tex]
what is trigonometry?The area of mathematics known as trigonometry examines the correlation between triangle side lengths and angles. The area first appeared in the Hellenistic era, around the third century BC. from the use of geometry in astronomical study. The area of mathematics known as exact methods deals with specific trigonometric functions and how they might be used in calculations. There are six popular trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their respective names and acronyms (csc). Studying the characteristics of triangles, particularly right triangles, is called trigonometry. The study of geometry, however, is the characteristics of all geometric figures.
[tex]sin^6 x - cos^6x\\(sin^3x)^2 - (cos^3x)^2\\(sin x - cosx)(sinx + cosx)(1 + sinx cosx)(1 - sinx cosx)\\(sin^2x - cos^2x )( 1- sin^2xcos^x)\\[/tex]
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The scatter plot shows a hiker's elevation above sea level during a hike from the base to the
top of a mountain. The equation of a trend line for the hiker's elevation is y=9.16x +659, where x
represents the number of minutes and y represents the hiker's elevation in feet. Use the equation of
the trend line to estimate the hiker's elevation after 150 minutes.
The trend line equation to estimate the hiker's elevation after 150 minutes is 2033 feet.
What is the hiker's elevation?Given that,
A trend line's equation for the hiker's elevation is y = 9.16x + 659.
We must determine,
The trend line equation is used to calculate the hiker's elevation after 150 minutes.
In response to the question,
A trend line's equation for the hiker's elevation is y = 9.16x + 659.
Where x is the number of minutes and y is the elevation of the hiker in feet.
Substitute the value of x = 150minute in the given equation to find the trend line to estimate the hiker's elevation after 150minutes.
Therefore,
y = 9.16x + 659
x = 150
y = 9.16(150) + 659
y = 1374 + 659
y = 2033
As a result, the trend line equation for estimating the hiker's elevation after 150 minutes is 2033 feet.
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Is (5, 0) a solution to the equation y = 3x2 + 5?
By replacing the values of the point in the quadratic equation, we can see that it is not a solution.
Is (5, 0) a solution for the given equation?Here we have the quadratic equation:
y = 3x^2 + 5
And we want to see if the point 3x^2 + 5 is a solution of the quadratic equation.
To check that, we need to replace the values of the point in the equation and see if the equation is true.
We need to replace:
x = 5
y = 0
We will get:
0 = 3*5^2 + 5
0 = 3*25 + 5
0 = 75 + 5
0 = 80
This is a false equation, then (5,0) is not a solution.
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What is midpoint formula Class 11?
The midpoint formula is a mathematical formula used to find the midpoint of a line segment, given the coordinates of the two endpoints of the segment. In class 11, it is defined as:
Midpoint = ( (x1 + x2)/2 , (y1 + y2)/2 )
Where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the line segment, and (x, y) are the coordinates of the midpoint of the segment.
For example, if the two endpoints of a line segment are (5,6) and (9,11)
Midpoint = ( (5 + 9)/2 , (6 + 11)/2 )
Midpoint = (7,8.5)
The midpoint of the line segment with endpoints (5,6) and (9,11) is (7,8.5)
It's worth noting that this formula can be adapted for three-dimensional points, where the midpoint formula would be
((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)
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Ade and Ola share a sum of 4000.00 in the ratio of 3:5 find the amount Ade received solutions
The number of shares that Ade and Ola possess will be 1500 and 2500, respectively.
What is the solution to the equation?The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.
Ade and Ola share a sum of 4000.00 in a ratio of 3:5.
Let 'x' and 'y' be the number of shares that Ade and Ola have, respectively. Then the equations are given as,
x / y = 3 / 5 ...1
x + y = 4000 ...2
From equations 1 and 2, then we have
(3/5)y + y = 4000
(8/5) y = 4000
y = 2500
Then the value of the variable 'x' is calculated as,
x + 2500 = 4000
x = 1500
The number of shares that Ade and Ola possess will be 1500 and 2500, respectively.
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How do you solve equations of equality?
Answer:
If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal. When you solve an equation, you find the value of the variable that makes the equation true. In order to solve the equation, you isolate the variable.
Step-by-step explanation:
( x - 8 )( x + 8)
this is Operation Multiplication btw
For the expression (x - 8)(x + 8), using the arithmetic operation the value for x is x = 8.
What is arithmetic operation?
A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.
The given expression is - (x - 8)(x + 8)
Use the arithmetic operation of multiplication -
(x - 8)(x + 8)
x(x + 8) - 8(x + 8)
x² + 8x - 8x -64
Now, use the arithmetic operation of addition and subtraction -
x² + 8x - 8x -64
x² - 64
Now, collect the like terms on each side -
x² = 64
x = √64
Now, solve the radical form to get the value of x -
x = √64
x = 8
Therefore, the value of x is obtained as x = 8.
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Answer!! What is Simplified?
On solving the provided question, we can say that Volume of cuboid = 2x(3x+4)(3x-4)
Volume = [tex]18x^3 + 48x^2 + 32x[/tex]
what is cuboid?A cuboid is a solid or three-dimensional form in geometry. A cuboid is a convex polyhedron having 8 vertices, 12 sides, and 6 rectangular faces. Another name for a cuboid is a cuboid. A cube is a cuboid with six square faces. Boxes include things like books and bricks. The following are the key variations between cubic and cubic: In contrast to a cube, which has rectangular faces, a cube has six equally sized square faces. Even though the cube and cuboid structures have a similar appearance, they differ in some ways in terms of side length, diagonal, and area.
here,
we have a cuboid
Volume of cuboid = 2x(3x+4)(3x-4)
Volume = [tex]18x^3 + 48x^2 + 32x[/tex]
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The perimeter (y) of a given shape is a function of its longest side (x). The equation for this function is y = 2.5x + 14. What does a solution of (15 feet, 51.5 feet) represent in the context of the problem?
The solution of (15 feet, 51.5 feet) represents the perimeter of a shape with a longest side of 15 feet.
What is Equation?Equation is a mathematical statement that express the equality of two expression use in equal sign.
The equation for this perimeter can be written as y = 2.5x + 14, where y is the perimeter of the shape and x is the longest side of the shape.
Thus, when x is 15 feet, the perimeter of the shape is 51.5 feet, as represented by the solution (15 feet, 51.5 feet).
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The dimensions of a rectangle are 7 in by (x+2)in. Which expression represents its area?
The correct option for the area of rectangle is A: 7(x + 2) sq in.
Define the term rectangle and its area?The characteristics of a rectangle were just as follows:
It has a tight, flat form.There are 4 corners, 4 angles, as well as 4 sides (vertices).There are two dimensions to it: length and width.A rectangle has 90° angles on each side.The opposing sides are parallel and equal.It features two equal-length diagonals.An area is the amount of space a rectangle takes up.By calculating the product of a rectangle's length and breadth, one can get the area of the shape.
So,
Area of a rectangle = Length × Width
Dimension are-
length : 7 inBreadth : (x + 2)inSo,
Area of a rectangle = 7 × (x + 2) sq in..
Thus, the expression represents area of a rectangle is 7 × (x + 2) sq in.
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The complete question is-
The dimensions of a rectangle are 7 in by (x+2)in. Which expression represents its area?
A: 7(x + 2) sq in..
B: 7 + (x + 2) sq in..
Which type of dilation are the given scale factors?
Select Expansion or Contraction to correctly describe the type of dilation for each given scale factor.
Scale Factor Expansion
336
554
0.5
-1.2
O
Contraction
O
O
O
O
ہے
Enlarging or contracting a figure's size is known as dilatation. In other words, a scale factor is used to either enlarge or diminish the image while maintaining similarity between the preimage and the image.
What Is A Dilation ?In line with the illustrations below. A dilation that produces a smaller image is called a reduction (conjure up shrinking), whereas a dilation that produces a larger image is called an enlargement (conjure up stretching).Image reduction occurs when the scale factor is between 0 and 1.An expansion of the image occurs when the scale factor exceeds 1.The center point of the dilation is located, and the distances between this center point and various points on the preimage and the image are measured in order to determine the scale factor for the dilation. According to Math Bits Notebook, we can determine the scale factor by dividing these distances by two.Given that each angle is congruent, each segment is proportional, the slope of each segment is preserved, and the perimeter of the preimage and image have the same scale factor, using dilation scale factors allows us to enlarge or reduce a figure to the desired size.To Learn more About dilatation Refer To:
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There were 75 runners to start a race. In the first half of the race, 1/3 of them dropped out. In the second half of the race, 2/5 of the remaining runners dropped out. How many runners finished the race?
What is the coefficient of the variable in the expression 2 − 5x − 4 8?
So, on solving the provided question, we can say that coefficient of the variable in the expression 2 − 5x − 4 8 is 5
What is the coefficient?A coefficient in mathematics is a polynomial, a series, or the multiplicative coefficient of a particular term in an expression. Typically numeric, however any expression is permitted. The term "parameter" can also refer to the coefficients themselves if they are variables. A number times a variable equals a coefficient. Coefficient examples include: The coefficient is 14 in phrase 14c 14c 14c. The coefficient is 1 for word g. multiplies the variable by this amount. example Given that "z" is a variable and that 6z is the definition of the term, the coefficient is 6. One is the coefficient of the square of x.
here,
coefficient of the variable in the expression 2 − 5x − 4 8 is 5
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when asked to find the prime factorization of 16 a student writes two times eight explain the student's error
The student's error is that 16 has to be prime factorized as (2)⁴ or 2 × 2 × 2 × 2.
What is Prime factorization?Prime factorization is defined as the method of writing a number as the product of two or more prime numbers.
Prime numbers are numbers which has only two factors, 1 and the number itself.
Given number is 16.
We have to factorize 16 as a product of prime factors.
Student wrote it as,
16 = 2 × 8
The error here is that 8 is not a prime number. We have to again factorize 8 to get the required factored form.
8 = 2 × 4
4 = 2 × 2
So combining all these,
16 = 2 × 2 × 2 × 2
So 16 can be prime factorized as 16 = 2 × 2 × 2 × 2.
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3/4 times (15/4-3 1/2) divided bye 1 1/3
Answer:
the awsner is 20 I believe so don't trust me or trust me i don't care
Step-by-step explanation:
I don't know
Answer:
9/64
Step-by-step explanation:
1 3/4 x (15/4-3 1/2) divided by 1 1/3
2 3/4x1/4 divided by 4/3
3 3/16 divided by 4/3
4 3/16x3/4=9/64
What does to the power of negative 1 mean?
positive number to the power negative 1 is a number that is always less than 1.
Explanation: a-m = 1/am. when calculate have the result when a positive number is raised to the power of -1, we always get a number that is less than one; for example, 2-1 = 1/2 < 1.
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