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

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

Options:

Correct Answer: 7

Solution:

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

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

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

Step 1: Identify the given quadratic equation, which is x² + 5x + k = 0.
Step 2: Recognize that the roots of the equation are given as 1 and 4.
Step 3: Use the product of the roots formula, which states that the product of the roots (1 * 4) equals k.
Step 4: Calculate the product: 1 * 4 = 4, so k = 4.
Step 5: Use the sum of the roots formula, which states that the sum of the roots (1 + 4) should equal -b/a, where b is the coefficient of x.
Step 6: Calculate the sum: 1 + 4 = 5, which matches the coefficient of x in the equation (5).
Step 7: Since both the product and sum conditions are satisfied, we confirm that k = 4.

Quadratic Equations – Understanding the relationship between the coefficients of a quadratic equation and its roots, specifically using the sum and product of roots.
Roots of Equations – Applying the known roots of a quadratic equation to find the unknown coefficient.