Q. Factor the polynomial x^2 + 7x + 10.
A.
(x + 5)(x + 2)
B.
(x + 10)(x - 1)
C.
(x - 5)(x - 2)
D.
(x + 1)(x + 10)
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Solution
To factor, we look for two numbers that multiply to 10 and add to 7. The numbers are 5 and 2. Thus, the factorization is (x + 5)(x + 2).
Correct Answer:
A
— (x + 5)(x + 2)
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Q. What are the roots of the equation 2x^2 + 3x - 2 = 0?
A.
x = 1, x = -2
B.
x = -1, x = 2
C.
x = 2, x = -1.5
D.
x = -2, x = 1.5
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Solution
Using the quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a, where a = 2, b = 3, c = -2, we find the roots to be x = 2 and x = -1.5.
Correct Answer:
C
— x = 2, x = -1.5
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Q. What are the roots of the equation x^2 + 4x + 4 = 0?
A.
x = -2, -2
B.
x = 2, 2
C.
x = -4, 0
D.
x = 0, 4
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Solution
The equation can be factored as (x + 2)(x + 2) = 0. Thus, the root is x = -2 with multiplicity 2.
Correct Answer:
A
— x = -2, -2
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Q. What are the roots of the equation x^2 + 6x + 9 = 0?
A.
-3 and -3
B.
3 and 3
C.
0 and 9
D.
1 and 8
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Solution
This is a perfect square: (x + 3)(x + 3) = 0. Thus, the root is x = -3 (with multiplicity 2).
Correct Answer:
A
— -3 and -3
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Q. What are the roots of the quadratic equation 2x^2 - 8x = 0?
A.
0 and 4
B.
2 and 4
C.
4 and 2
D.
0 and 2
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Solution
Factoring gives 2x(x - 4) = 0. Thus, the roots are x = 0 and x = 4.
Correct Answer:
A
— 0 and 4
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Q. What are the solutions to the equation x^2 - 6x + 9 = 0?
A.
x = 3, 3
B.
x = -3, -3
C.
x = 6, 0
D.
x = 0, 6
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Solution
The equation can be factored as (x - 3)(x - 3) = 0. Thus, the solution is x = 3 with multiplicity 2.
Correct Answer:
A
— x = 3, 3
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Q. What is the solution to the equation 2x^2 - 8 = 0?
A.
x = 2, -2
B.
x = 4, -4
C.
x = 0, 4
D.
x = 4, 0
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Solution
First, add 8 to both sides: 2x^2 = 8. Then divide by 2: x^2 = 4. Taking the square root gives x = ±4.
Correct Answer:
B
— x = 4, -4
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Q. What is the value of x in the equation 3x + 4 = 10?
A.
x = 2
B.
x = 3
C.
x = 4
D.
x = 1
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Solution
To solve 3x + 4 = 10, subtract 4 from both sides: 3x = 6. Then divide by 3: x = 2.
Correct Answer:
A
— x = 2
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Q. What is the vertex of the quadratic function y = 2x^2 - 8x + 5?
A.
(2, -3)
B.
(4, -3)
C.
(2, 5)
D.
(4, 5)
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Solution
The vertex can be found using the formula x = -b/(2a). Here, a = 2 and b = -8, so x = 8/4 = 2. Plugging x = 2 into the equation gives y = 2(2)^2 - 8(2) + 5 = -3. Thus, the vertex is (2, -3).
Correct Answer:
A
— (2, -3)
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Q. Which of the following is a factor of the polynomial x^2 - 4?
A.
x - 2
B.
x + 2
C.
x^2 + 4
D.
x - 1
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Solution
The polynomial x^2 - 4 can be factored as (x - 2)(x + 2). Therefore, x - 2 is a factor.
Correct Answer:
A
— x - 2
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