Quadratic Equations

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Q1. 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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Q2. Determine the product of the roots of the equation x² + 6x + 8 = 0. (2023)

Solution:

The product of the roots is given by c/a = 8/1 = 8.

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Q3. If the roots of the equation x² + px + 12 = 0 are 3 and 4, what is the value of p? (2020)

Solution:

The sum of the roots is -p = 3 + 4 = 7, hence p = -7.

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Q4. 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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Q5. 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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Q6. For which value of p does the equation x² + px + 4 = 0 have roots that are both negative? (2022)

Solution:

For both roots to be negative, p must be greater than 0 and p² > 16. Thus, p < -4.

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Q7. What are the roots of the equation x² - 5x + 6 = 0? (2021)

Solution:

The roots can be found using the factorization method: (x - 2)(x - 3) = 0, hence the roots are 2 and 3.

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Q8. For which value of k does the equation x² - kx + 9 = 0 have no real roots? (2021)

Solution:

The discriminant must be less than zero: k² - 4*1*9 < 0, thus k² < 36, so k > 6 or k < -6.

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Q9. Determine the roots of the equation x² + 2x - 8 = 0. (2023)

Solution:

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

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Q10. What is the value of k if the equation x² - 4x + k = 0 has no real roots? (2021)

Solution:

For no real roots, the discriminant must be less than zero: (-4)² - 4*1*k < 0, hence k > 4.