Linear and Quadratic Equations
Q. Find the value of x in the equation 3x^2 - 12 = 0.
Solution
Add 12 to both sides: 3x^2 = 12. Divide by 3: x^2 = 4. Thus, x = ±2.
Correct Answer: B — 2
Q. Find the value of x in the equation 4x^2 + 8x + 3 = 0.
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A.
x = -1
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B.
x = -3/2
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C.
x = -1/2
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D.
x = -3
Solution
Using the quadratic formula x = [-b ± √(b² - 4ac)] / 2a gives x = [-8 ± √(64 - 48)] / 8 = [-8 ± 4] / 8.
Correct Answer: B — x = -3/2
Q. Find the value of x in the equation 4x^2 - 16x + 15 = 0.
Solution
Factoring gives (4x - 3)(x - 5) = 0, so x = 3 or x = 5.
Correct Answer: B — 3
Q. Find the value of x in the equation 5x - 2 = 3x + 6.
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A.
x = 2
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B.
x = 3
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C.
x = 4
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D.
x = 5
Solution
Subtract 3x from both sides: 2x - 2 = 6. Add 2: 2x = 8. Divide by 2: x = 4.
Correct Answer: A — x = 2
Q. Find the value of x in the equation 5x - 2(3 - x) = 4.
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A.
x = 1
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B.
x = 2
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C.
x = 3
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D.
x = 4
Solution
Distributing gives 5x - 6 + 2x = 4. Combining like terms: 7x - 6 = 4. Thus, 7x = 10, x = 10/7.
Correct Answer: B — x = 2
Q. Find the value of x in the equation x^2 + 6x + 9 = 0.
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A.
x = -3
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B.
x = 3
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C.
x = 0
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D.
x = -9
Solution
This is a perfect square: (x + 3)^2 = 0, so x = -3.
Correct Answer: A — x = -3
Q. If 2x^2 + 3x - 5 = 0, what is the value of x using the quadratic formula?
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A.
x = 1
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B.
x = -1
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C.
x = 2
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D.
x = -2
Solution
Using the quadratic formula x = [-b ± √(b² - 4ac)] / 2a, we find x = [-3 ± √(9 + 40)] / 4.
Correct Answer: B — x = -1
Q. If 2x^2 + 8x + 6 = 0, what is the value of x?
-
A.
x = -1
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B.
x = -3
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C.
x = -2
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D.
x = -4
Solution
Dividing the equation by 2 gives x^2 + 4x + 3 = 0, which factors to (x + 1)(x + 3) = 0.
Correct Answer: B — x = -3
Q. If 4x^2 - 16 = 0, what is the value of x?
-
A.
x = 2
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B.
x = -2
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C.
x = 4
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D.
x = -4
Solution
Add 16 to both sides: 4x^2 = 16. Divide by 4: x^2 = 4. Thus, x = ±2.
Correct Answer: A — x = 2
Q. If 5x - 2 = 3x + 6, what is the value of x?
-
A.
x = 2
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B.
x = 3
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C.
x = 4
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D.
x = 5
Solution
Subtract 3x from both sides: 2x - 2 = 6. Add 2: 2x = 8. Divide by 2: x = 4.
Correct Answer: B — x = 3
Q. If 5x^2 - 20 = 0, what is the value of x?
Solution
Add 20 to both sides: 5x^2 = 20. Divide by 5: x^2 = 4. Thus, x = ±2.
Correct Answer: B — 2
Q. If the roots of the equation x^2 + px + q = 0 are 3 and -2, what is the value of p?
Solution
Using the sum of roots: p = -(3 + (-2)) = -1.
Correct Answer: B — 5
Q. If the roots of the equation x^2 - px + q = 0 are 3 and 4, what is the value of p?
Solution
The sum of the roots is p = 3 + 4 = 7.
Correct Answer: A — 7
Q. If x^2 + 4x + 4 = 0, what is the value of x?
Solution
This is a perfect square: (x + 2)^2 = 0, so x = -2.
Correct Answer: A — -2
Q. If x^2 - 5x + 6 = 0, what are the roots of the equation?
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A.
x = 1, 6
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B.
x = 2, 3
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C.
x = -2, -3
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D.
x = 0, 6
Solution
Factoring gives (x - 2)(x - 3) = 0, so x = 2 and x = 3.
Correct Answer: B — x = 2, 3
Q. Solve for x: 2x + 3 = 11
Solution
Subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4.
Correct Answer: B — 3
Q. Solve for x: 3x^2 - 12 = 0.
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A.
x = 2
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B.
x = -2
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C.
x = 4
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D.
x = -4
Solution
Add 12 to both sides: 3x^2 = 12. Divide by 3: x^2 = 4. Thus, x = ±2.
Correct Answer: A — x = 2
Q. Solve for x: 4x^2 - 12x + 9 = 0.
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A.
x = 1
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B.
x = 3
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C.
x = 2
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D.
x = 4
Solution
This factors to (2x - 3)(2x - 3) = 0, giving the double root x = 3.
Correct Answer: B — x = 3
Q. Solve for x: 4x^2 - 16 = 0.
-
A.
x = 2
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B.
x = -2
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C.
x = 4
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D.
x = -4
Solution
Add 16 to both sides: 4x^2 = 16. Divide by 4: x^2 = 4. Thus, x = ±2.
Correct Answer: A — x = 2
Q. Solve for x: 5x^2 + 10x = 0.
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A.
x = 0, -2
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B.
x = 2, 0
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C.
x = -2, 2
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D.
x = 5, 0
Solution
Factoring gives 5x(x + 2) = 0, so x = 0 or x = -2.
Correct Answer: A — x = 0, -2
Q. Solve for x: x^2 + 6x + 9 = 0.
-
A.
x = -3
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B.
x = 3
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C.
x = 0
-
D.
x = -9
Solution
This factors to (x + 3)(x + 3) = 0, so x = -3.
Correct Answer: A — x = -3
Q. What are the roots of the equation x^2 - 5x + 6 = 0?
-
A.
1 and 6
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B.
2 and 3
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C.
3 and 4
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D.
0 and 6
Solution
Factoring gives (x - 2)(x - 3) = 0, so x = 2 and x = 3.
Correct Answer: B — 2 and 3
Q. What are the solutions to the equation x^2 + 2x - 8 = 0?
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A.
x = 2, -4
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B.
x = -2, 4
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C.
x = 4, -2
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D.
x = -4, 2
Solution
Factoring gives (x + 4)(x - 2) = 0, so x = -4 and x = 2.
Correct Answer: C — x = 4, -2
Q. What are the solutions to the equation x^2 + 4x + 4 = 0?
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A.
x = -2
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B.
x = 2
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C.
x = 0
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D.
x = -4
Solution
This factors to (x + 2)(x + 2) = 0, so the double root is x = -2.
Correct Answer: A — x = -2
Q. What is the discriminant of the equation 2x^2 + 3x + 1 = 0?
Solution
The discriminant is b^2 - 4ac = 3^2 - 4(2)(1) = 9 - 8 = 1.
Correct Answer: A — 1
Q. What is the discriminant of the equation 2x^2 - 4x + 2 = 0?
Solution
Discriminant D = b² - 4ac = (-4)² - 4(2)(2) = 16 - 16 = 0.
Correct Answer: A — 0
Q. What is the discriminant of the equation 3x^2 + 6x + 2 = 0?
Solution
The discriminant is b^2 - 4ac = 6^2 - 4*3*2 = 36 - 24 = 12.
Correct Answer: A — 0
Q. What is the product of the roots of the equation x^2 + 3x - 10 = 0?
Solution
The product of the roots is c/a = -10/1 = -10.
Correct Answer: A — -10
Q. What is the product of the roots of the equation x^2 + 5x + 6 = 0?
Solution
The product of the roots is c/a = 6/1 = 6.
Correct Answer: A — 6
Q. What is the product of the roots of the equation x^2 - 7x + 10 = 0?
Solution
The product of the roots is given by c/a = 10/1 = 10.
Correct Answer: A — 10
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