Q. Find the value of k for which the equation x^2 + kx + 9 = 0 has one real solution.
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Solution
For the equation to have one real solution, the discriminant must be zero: k^2 - 4*1*9 = 0. Thus, k^2 = 36, giving k = ±6. The correct answer is -9.
Correct Answer:
B
— -9
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Q. Find the value of x in the equation x^2 - 9 = 0.
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Solution
Factoring gives (x - 3)(x + 3) = 0. Thus, the solutions are x = 3 and x = -3.
Correct Answer:
A
— 3
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Q. Find the x-intercepts of the equation y = x^2 - 4.
A.
x = 2, -2
B.
x = 4, -4
C.
x = 0, 4
D.
x = -4, 0
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Solution
Set y = 0: 0 = x^2 - 4. This factors to (x - 2)(x + 2) = 0, giving x = 2 and x = -2.
Correct Answer:
A
— x = 2, -2
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Q. If one root of the equation x^2 + px + 6 = 0 is 2, what is the value of p?
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Solution
If one root is 2, then the other root can be found using the product of the roots: 2 * r = 6, so r = 3. The sum of the roots is 2 + 3 = -p, thus p = -5.
Correct Answer:
B
— -4
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Q. If one root of the equation x^2 + px + q = 0 is 3, what is the value of p if the other root is 1?
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Solution
Using the sum of the roots, p = -(3 + 1) = -4. The product of the roots gives q = 3 * 1 = 3.
Correct Answer:
B
— -6
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Q. Solve for x: x^2 + 2x - 8 = 0.
A.
x = 2, -4
B.
x = -2, 4
C.
x = 4, -2
D.
x = -4, 2
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Solution
Factoring gives (x + 4)(x - 2) = 0. Thus, the solutions are x = -4 and x = 2.
Correct Answer:
C
— x = 4, -2
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Q. Solve the equation 2x^2 + 4x - 6 = 0.
A.
x = 1, -3
B.
x = -1, 3
C.
x = 3, -1
D.
x = -3, 1
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Solution
First, divide the equation by 2: x^2 + 2x - 3 = 0. Factor to (x + 3)(x - 1) = 0. Thus, x = -3 and x = 1.
Correct Answer:
A
— x = 1, -3
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Q. Solve the equation x^2 + 4x + 4 = 0.
A.
x = -2
B.
x = 2
C.
x = -4
D.
x = 0
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Solution
This can be factored as (x + 2)(x + 2) = 0. Therefore, x = -2 is a double root.
Correct Answer:
A
— x = -2
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Q. Solve the quadratic equation x^2 + 4x + 4 = 0.
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Solution
This can be factored as (x + 2)(x + 2) = 0. Therefore, x = -2 is a double root.
Correct Answer:
A
— -2
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Q. What are the solutions to the equation x^2 - 10x + 25 = 0?
A.
x = 5
B.
x = 0, 5
C.
x = 10
D.
x = 2, 3
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Solution
The equation can be factored as (x - 5)(x - 5) = 0. Thus, the solution is x = 5 (a repeated root).
Correct Answer:
A
— x = 5
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Q. What is the discriminant of the quadratic equation 2x^2 + 3x + 1 = 0?
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Solution
The discriminant is calculated as b^2 - 4ac. Here, a = 2, b = 3, c = 1. So, discriminant = 3^2 - 4(2)(1) = 9 - 8 = 1.
Correct Answer:
A
— 1
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Q. What is the discriminant of the quadratic equation 2x^2 + 4x + 2 = 0?
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Solution
The discriminant is given by b^2 - 4ac. Here, a = 2, b = 4, c = 2. So, the discriminant = 4^2 - 4(2)(2) = 16 - 16 = 0.
Correct Answer:
A
— 0
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Q. What is the discriminant of the quadratic equation 3x^2 + 6x + 2 = 0?
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Solution
The discriminant is given by b^2 - 4ac. Here, a = 3, b = 6, c = 2. Thus, the discriminant = 6^2 - 4*3*2 = 36 - 24 = 12.
Correct Answer:
A
— 0
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Q. What is the discriminant of the quadratic equation 3x^2 + 6x + 2?
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Solution
The discriminant is given by b^2 - 4ac. Here, a = 3, b = 6, c = 2. Thus, discriminant = 6^2 - 4(3)(2) = 36 - 24 = 12.
Correct Answer:
B
— 4
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Q. What is the factored form of the quadratic equation x^2 + 6x + 9?
A.
(x + 3)(x + 3)
B.
(x - 3)(x - 3)
C.
(x + 9)(x + 1)
D.
(x - 9)(x - 1)
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Solution
This quadratic can be factored as (x + 3)(x + 3) or (x + 3)^2.
Correct Answer:
A
— (x + 3)(x + 3)
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Q. What is the factored form of the quadratic expression x^2 - 7x + 10?
A.
(x - 5)(x - 2)
B.
(x - 10)(x + 1)
C.
(x - 1)(x - 10)
D.
(x - 2)(x - 5)
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Solution
To factor, we look for two numbers that multiply to 10 and add to -7. The numbers -2 and -5 work, so the factored form is (x - 2)(x - 5).
Correct Answer:
D
— (x - 2)(x - 5)
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Q. What is the factored form of the quadratic expression x^2 - 9?
A.
(x - 3)(x + 3)
B.
(x - 9)(x + 1)
C.
(x - 1)(x + 1)
D.
(x + 3)(x + 3)
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Solution
The expression x^2 - 9 is a difference of squares, which factors to (x - 3)(x + 3).
Correct Answer:
A
— (x - 3)(x + 3)
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Q. What is the sum of the roots of the equation x^2 - 8x + 15 = 0?
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Solution
The sum of the roots can be found using -b/a. Here, b = -8 and a = 1. Thus, the sum is 8.
Correct Answer:
A
— 8
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Q. What is the sum of the roots of the quadratic equation x^2 + 3x - 10 = 0?
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Solution
The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is given by -b/a. Here, b = 3 and a = 1, so the sum is -3/1 = -3.
Correct Answer:
A
— -3
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Q. Which of the following is a solution to the equation 2x^2 - 8 = 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 = ±2. Thus, 2 is a solution.
Correct Answer:
B
— 2
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Q. Which of the following is a solution to the equation x^2 + 6x + 9 = 0?
A.
x = -3
B.
x = 3
C.
x = 0
D.
x = -6
Show solution
Solution
The equation can be factored as (x + 3)(x + 3) = 0. Thus, the solution is x = -3.
Correct Answer:
A
— x = -3
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