What type of equation will best fit the data below?
a coordinate plane with points plotted that form a U shape.
Responses
linear
quadratic
exponential
11 answers
The best type of equation that would fit the data described, which forms a U shape, is a quadratic equation.
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Use the graph of f (x) to find the solutions to the equation f (x) = 0.
A coordinate plane with parabola opening up. The y-intercept is negative 12 and the x-intercepts are 2 and negative 6.
(1 point)
Responses
two solutions: x = 6, negative 2
two solutions: x = Image with alt text: 6, negative 2
two solutions: x = negative 6, 2
two solutions: x = Image with alt text: negative 6, 2
one solution: x = negative 12
one solution: x = Image with alt text: negative 12
no solutions
A coordinate plane with parabola opening up. The y-intercept is negative 12 and the x-intercepts are 2 and negative 6.
(1 point)
Responses
two solutions: x = 6, negative 2
two solutions: x = Image with alt text: 6, negative 2
two solutions: x = negative 6, 2
two solutions: x = Image with alt text: negative 6, 2
one solution: x = negative 12
one solution: x = Image with alt text: negative 12
no solutions
We can see from the description that the parabola opens up, and it has x-intercepts at 2 and negative 6. Therefore, the solutions to the equation f(x) = 0 are two solutions: x = 2 and x = -6. Thus, the correct response is: two solutions: x = 2, -6.
What are the solutions of the equation 2 x squared equals 2? Use a graph of a related function whose roots answer the question.
(1 point)
Responses
An upward facing parabola passes through points left parenthesis negative 1 comma 0 right parenthesis, left parenthesis 0 comma negative 1 right parenthesis, and left parenthesis 1 comma 0 right parenthesis.
There are two solutions: x equals plus or minus 1
Image with alt text: An upward facing parabola passes through points left parenthesis negative 1 comma 0 right parenthesis, left parenthesis 0 comma negative 1 right parenthesis, and left parenthesis 1 comma 0 right parenthesis. There are two solutions: Image with alt text: x equals plus or minus 1
An upward facing parabola passes through points left parenthesis negative 2 comma 0 right parenthesis, left parenthesis 0 comma negative 2 right parenthesis, and left parenthesis 2 comma 0 right parenthesis.
There are two solutions: x equals plus or minus 2
Image with alt text: An upward facing parabola passes through points left parenthesis negative 2 comma 0 right parenthesis, left parenthesis 0 comma negative 2 right parenthesis, and left parenthesis 2 comma 0 right parenthesis. There are two solutions: Image with alt text: x equals plus or minus 2
An upward facing parabola passes through points left parenthesis negative 2 comma 5 right parenthesis, left parenthesis 0 comma 1 right parenthesis, and left parenthesis 2 comma 5 right parenthesis.
There are no solutions.
Image with alt text: An upward facing parabola passes through points left parenthesis negative 2 comma 5 right parenthesis, left parenthesis 0 comma 1 right parenthesis, and left parenthesis 2 comma 5 right parenthesis. There are no solutions.
An upward facing parabola passes through points left parenthesis negative 2 comma 2 right parenthesis, left parenthesis 0 comma negative 2 right parenthesis, and left parenthesis 2 comma 2 right parenthesis.
There are two solutions: x equals plus or minus start root 2 end root
Image with alt text: An upward facing parabola passes through points left parenthesis negative 2 comma 2 right parenthesis, left parenthesis 0 comma negative 2 right parenthesis, and left parenthesis 2 comma 2 right parenthesis. There are two solutions: Image with alt text: x equals plus or minus start root 2 end root
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(1 point)
Responses
An upward facing parabola passes through points left parenthesis negative 1 comma 0 right parenthesis, left parenthesis 0 comma negative 1 right parenthesis, and left parenthesis 1 comma 0 right parenthesis.
There are two solutions: x equals plus or minus 1
Image with alt text: An upward facing parabola passes through points left parenthesis negative 1 comma 0 right parenthesis, left parenthesis 0 comma negative 1 right parenthesis, and left parenthesis 1 comma 0 right parenthesis. There are two solutions: Image with alt text: x equals plus or minus 1
An upward facing parabola passes through points left parenthesis negative 2 comma 0 right parenthesis, left parenthesis 0 comma negative 2 right parenthesis, and left parenthesis 2 comma 0 right parenthesis.
There are two solutions: x equals plus or minus 2
Image with alt text: An upward facing parabola passes through points left parenthesis negative 2 comma 0 right parenthesis, left parenthesis 0 comma negative 2 right parenthesis, and left parenthesis 2 comma 0 right parenthesis. There are two solutions: Image with alt text: x equals plus or minus 2
An upward facing parabola passes through points left parenthesis negative 2 comma 5 right parenthesis, left parenthesis 0 comma 1 right parenthesis, and left parenthesis 2 comma 5 right parenthesis.
There are no solutions.
Image with alt text: An upward facing parabola passes through points left parenthesis negative 2 comma 5 right parenthesis, left parenthesis 0 comma 1 right parenthesis, and left parenthesis 2 comma 5 right parenthesis. There are no solutions.
An upward facing parabola passes through points left parenthesis negative 2 comma 2 right parenthesis, left parenthesis 0 comma negative 2 right parenthesis, and left parenthesis 2 comma 2 right parenthesis.
There are two solutions: x equals plus or minus start root 2 end root
Image with alt text: An upward facing parabola passes through points left parenthesis negative 2 comma 2 right parenthesis, left parenthesis 0 comma negative 2 right parenthesis, and left parenthesis 2 comma 2 right parenthesis. There are two solutions: Image with alt text: x equals plus or minus start root 2 end root
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To find the solutions of the equation 2x^2 = 2, we can graph the related function y = 2x^2 - 2 and identify the x-intercepts.
The response that correctly describes the graph of the related function and the solutions is:
An upward facing parabola passes through points (-1, 0), (0, -1), and (1, 0).
There are two solutions: x = -1 and x = 1.
The response that correctly describes the graph of the related function and the solutions is:
An upward facing parabola passes through points (-1, 0), (0, -1), and (1, 0).
There are two solutions: x = -1 and x = 1.
Solve the equation by completing the square.
x squared plus 9 x minus 14 equals 0
(1 point)
Responses
10.35, 1.35
10.35, 1.35
10.35, negative 1 point 35
10.35, Image with alt text: negative 1 point 35
negative 10.35, negative 1.35
Image with alt text: negative 10.35, negative 1.35
negative 10.35, 1.35
Image with alt text: negative 10.35 , 1.35
x squared plus 9 x minus 14 equals 0
(1 point)
Responses
10.35, 1.35
10.35, 1.35
10.35, negative 1 point 35
10.35, Image with alt text: negative 1 point 35
negative 10.35, negative 1.35
Image with alt text: negative 10.35, negative 1.35
negative 10.35, 1.35
Image with alt text: negative 10.35 , 1.35
To solve the equation x^2 + 9x - 14 = 0 by completing the square, we follow these steps:
1. Move the constant term to the right side of the equation:
x^2 + 9x = 14
2. Take half of the coefficient of x (which is 9) and square it. Add this result to both sides of the equation:
x^2 + 9x + (9/2)^2 = 14 + (9/2)^2
x^2 + 9x + 81/4 = 14 + 81/4
3. Simplify both sides of the equation:
x^2 + 9x + 81/4 = 98/4 + 81/4
x^2 + 9x + 81/4 = 179/4
4. Rewrite the left side of the equation as a perfect square:
(x + 9/2)^2 = 179/4
5. Take the square root of both sides, considering both positive and negative square roots:
x + 9/2 = ±√(179/4)
6. Solve for x by isolating it on the left side:
x = -9/2 ± √(179/4)
The correct response is: x = -9/2 ± √(179/4), which can be approximated to x ≈ -10.35 and x ≈ 1.35.
1. Move the constant term to the right side of the equation:
x^2 + 9x = 14
2. Take half of the coefficient of x (which is 9) and square it. Add this result to both sides of the equation:
x^2 + 9x + (9/2)^2 = 14 + (9/2)^2
x^2 + 9x + 81/4 = 14 + 81/4
3. Simplify both sides of the equation:
x^2 + 9x + 81/4 = 98/4 + 81/4
x^2 + 9x + 81/4 = 179/4
4. Rewrite the left side of the equation as a perfect square:
(x + 9/2)^2 = 179/4
5. Take the square root of both sides, considering both positive and negative square roots:
x + 9/2 = ±√(179/4)
6. Solve for x by isolating it on the left side:
x = -9/2 ± √(179/4)
The correct response is: x = -9/2 ± √(179/4), which can be approximated to x ≈ -10.35 and x ≈ 1.35.
What is the simplified form of The quantity x to the fourth power minus 81 divided by the quantity x plus 3 ?
(1 point)
Responses
x cubed minus 3 times x squared plus 9 times x minus 27
Image with alt text: x cubed minus 3 times x squared plus 9 times x minus 27
the quantity x squared plus 9 times the quantity x squared minus 9
Image with alt text: the quantity x squared plus 9 times the quantity x squared minus 9
x cubed plus 3 times x squared plus 9 times x plus 27
Image with alt text: x cubed plus 3 times x squared plus 9 times x plus 27
x cubed plus 3 times x squared minus 9 times x minus 27
Image with alt text: x cubed plus 3 times x squared minus 9 times x minus 27
(1 point)
Responses
x cubed minus 3 times x squared plus 9 times x minus 27
Image with alt text: x cubed minus 3 times x squared plus 9 times x minus 27
the quantity x squared plus 9 times the quantity x squared minus 9
Image with alt text: the quantity x squared plus 9 times the quantity x squared minus 9
x cubed plus 3 times x squared plus 9 times x plus 27
Image with alt text: x cubed plus 3 times x squared plus 9 times x plus 27
x cubed plus 3 times x squared minus 9 times x minus 27
Image with alt text: x cubed plus 3 times x squared minus 9 times x minus 27
To simplify the expression (x^4 - 81)/(x + 3), we can factor the numerator as the difference of squares:
(x^4 - 81) = (x^2)^2 - 9^2
= (x^2 - 9)(x^2 + 9)
Therefore, the simplified form of the expression is:
[(x^2 - 9)(x^2 + 9)] / (x + 3)
The correct response is: the quantity x squared minus 9 times the quantity x squared plus 9.
(x^4 - 81) = (x^2)^2 - 9^2
= (x^2 - 9)(x^2 + 9)
Therefore, the simplified form of the expression is:
[(x^2 - 9)(x^2 + 9)] / (x + 3)
The correct response is: the quantity x squared minus 9 times the quantity x squared plus 9.
1 answer