Q1. What is the solution to the equation 2x^2 - 4x - 6 = 0 using the quadratic formula?
Solution:
Using the quadratic formula x = (-b ± √(b² - 4ac)) / 2a. Here, a = 2, b = -4, c = -6. Discriminant = (-4)² - 4(2)(-6) = 16 + 48 = 64. Thus, x = (4 ± √64) / 4 = (4 ± 8) / 4. Solutions are x = 3 and x = -1.
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Q2. What is the solution to the inequality 2x - 3 < 5?
Solution:
Add 3 to both sides: 2x < 8. Divide by 2: x < 4.
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Q3. Find the roots of the equation 3x^2 - 12 = 0.
Solution:
Add 12 to both sides: 3x^2 = 12. Divide by 3: x^2 = 4. Thus, x = ±2.
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Q4. Solve for x in the equation x^2 + 4x + 4 = 0.
Solution:
This factors to (x + 2)(x + 2) = 0. Therefore, x = -2.
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Q5. If 2x + 3 > 7, what is the solution for x?
Solution:
Subtract 3 from both sides: 2x > 4. Divide by 2: x > 2.
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Q6. Solve for x: 4x^2 + 8x = 0.
Solution:
Factor out 4x: 4x(x + 2) = 0. Thus, x = 0 or x + 2 = 0 gives x = -2.
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Q7. What are the solutions to the equation x^2 - 9 = 0?
Solution:
Factoring gives (x - 3)(x + 3) = 0. Thus, the solutions are x = 3 and x = -3.
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Q8. Find the roots of the quadratic equation 3x^2 + 6x + 3 = 0.
Solution:
Dividing the equation by 3 gives x^2 + 2x + 1 = 0, which factors to (x + 1)(x + 1) = 0. Thus, x = -1.
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Q9. What are the solutions to the equation x^2 - 10x + 21 = 0?
Solution:
Factoring gives (x - 3)(x - 7) = 0. Thus, the solutions are x = 3 and x = 7.
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Q10. Find the roots of the quadratic equation 4x^2 - 12x + 9 = 0.
Solution:
This can be factored as (2x - 3)(2x - 3) = 0. Thus, x = 3.
