Q1. What is the discriminant of the quadratic equation 2x^2 + 4x + 2 = 0?
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.
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Q2. Which of the following is a solution to the equation 2x^2 - 8 = 0?
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.
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Q3. Which of the following is a solution to the equation x^2 + 6x + 9 = 0?
Solution:
The equation can be factored as (x + 3)(x + 3) = 0. Thus, the solution is x = -3.
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Q4. 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?
Solution:
Using the sum of the roots, p = -(3 + 1) = -4. The product of the roots gives q = 3 * 1 = 3.
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Q5. Find the value of x in 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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Q6. If one root of the equation x^2 + px + 6 = 0 is 2, what is the value of p?
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.
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Q7. Find the value of k for which the equation x^2 + kx + 9 = 0 has one real solution.
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.
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Q8. What is the factored form of the quadratic expression x^2 - 9?
Solution:
The expression x^2 - 9 is a difference of squares, which factors to (x - 3)(x + 3).
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Q9. What is the discriminant of the quadratic equation 3x^2 + 6x + 2 = 0?
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.
