Quadratic Equations

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Q1. Solve the equation 2x^2 + 4x - 6 = 0.

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.

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Q2. Solve for x: x^2 + 2x - 8 = 0.

Solution:

Factoring gives (x + 4)(x - 2) = 0. Thus, the solutions are x = -4 and x = 2.

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Q3. What is the sum of the roots of the equation x^2 - 8x + 15 = 0?

Solution:

The sum of the roots can be found using -b/a. Here, b = -8 and a = 1. Thus, the sum is 8.

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Q4. What is the discriminant of the quadratic equation 3x^2 + 6x + 2?

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.

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Q5. Solve the equation x^2 + 4x + 4 = 0.

Solution:

This can be factored as (x + 2)(x + 2) = 0. Therefore, x = -2 is a double root.

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Q6. 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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Q7. Find the x-intercepts of the equation y = x^2 - 4.

Solution:

Set y = 0: 0 = x^2 - 4. This factors to (x - 2)(x + 2) = 0, giving x = 2 and x = -2.

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Q8. 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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Q9. Solve the quadratic equation x^2 + 4x + 4 = 0.

Solution:

This can be factored as (x + 2)(x + 2) = 0. Therefore, x = -2 is a double root.

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Q10. What is the factored form of the quadratic expression x^2 - 7x + 10?

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).