Quadratic Equations

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Q1. What is the value of the discriminant for the equation x² - 4x + 4 = 0? (2022)

Solution:

The discriminant is b² - 4ac = (-4)² - 4*1*4 = 0.

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Q2. The roots of the equation x² + 4x + k = 0 are 2 and -6. What is the value of k? (2021)

Solution:

Using the product of roots: k = 2 * (-6) = -12. The sum is 2 + (-6) = -4, which matches.

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Q3. Find the roots of the equation 4x² - 12x + 9 = 0. (2023)

Solution:

This is a perfect square: (2x - 3)² = 0, hence the root is x = 1.5.

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Q4. If one root of the equation x² - 7x + k = 0 is 3, what is the value of k? (2020)

Solution:

Using the root, substitute x = 3: 3² - 7*3 + k = 0, which gives k = 9.

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Q5. If the roots of the equation x² + 7x + p = 0 are -3 and -4, find p. (2019)

Solution:

Using the sum of roots (-3 + -4 = -7) and product of roots (-3*-4 = 12), we find p = 12.

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Q6. If the roots of the equation x² + mx + n = 0 are 1 and -1, find m and n. (2020)

Solution:

The sum of the roots is 0 (m = 0) and the product is -1 (n = 1).

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Q7. If the roots of the equation x² + 5x + k = 0 are 1 and 4, find k. (2020)

Solution:

Using the sum and product of roots: k = 1*4 = 4, and sum = 1 + 4 = 5, thus k = 7.

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Q8. For which value of k does the equation x² - kx + 9 = 0 have roots that are both positive? (2023)

Solution:

For both roots to be positive, k must be greater than 0 and k² > 4*9 = 36, thus k > 6.

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Q9. If one root of the equation x² - 7x + p = 0 is 3, what is the value of p? (2020)

Solution:

Using the root, substitute x = 3: 3² - 7*3 + p = 0, which gives p = 6.

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Q10. If one root of the equation x² - 6x + k = 0 is 2, find k. (2022)

Solution:

Using the root 2 in the equation: 2² - 6*2 + k = 0, we find k = 10.