Quadratic Equations

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Q1. If the roots of the equation x² + 5x + q = 0 are 1 and 4, find q. (2019)

Solution:

Using the product of roots: q = 1 * 4 = 4.

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Q2. What is the discriminant of the equation 3x² - 12x + 12 = 0? (2023)

Solution:

The discriminant is b² - 4ac = (-12)² - 4*3*12 = 144 - 144 = 0.

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Q3. 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.

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Q4. 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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Q5. Which of the following equations has roots that are both negative? (2022)

Solution:

The equation x² + 4x + 4 = 0 has roots -2 and -2, which are both negative.

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Q6. The roots of the equation x² + 2x + k = 0 are real and distinct if k is: (2020)

Solution:

For real and distinct roots, the discriminant must be positive: 2² - 4*1*k > 0, which simplifies to k < 1.

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Q7. If the roots of the equation x² - 6x + p = 0 are 2 and 4, what is the value of p? (2023)

Solution:

The product of the roots gives p = 2 * 4 = 8.

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Q8. What is the product of the roots of the equation x² - 8x + 15 = 0? (2022)

Solution:

The product of the roots is given by c/a = 15/1 = 15.

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Q9. Find the roots of the equation 3x² - 12x + 12 = 0. (2021)

Solution:

Dividing by 3 gives x² - 4x + 4 = 0, which factors to (x - 2)² = 0, hence the root is 2.

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Q10. What is the vertex of the parabola represented by the equation y = x² - 4x + 3? (2022)

Solution:

The vertex can be found using the formula x = -b/2a. Here, x = 2, and substituting back gives y = -1.