If the roots of the equation x² + 7x + k = 0 are -3 and -4, find k. (2022)

If the roots of the equation x² + 7x + k = 0 are -3 and -4, find k. (2022)

Step 1: Identify the given quadratic equation, which is x² + 7x + k = 0.
Step 2: Recognize that the roots of the equation are given as -3 and -4.
Step 3: Use the formula for the sum of the roots, which is -b/a. Here, b = 7 and a = 1, so the sum of the roots should equal -7.
Step 4: Calculate the sum of the given roots: -3 + -4 = -7. This matches the expected sum.
Step 5: Use the formula for the product of the roots, which is c/a. Here, c = k and a = 1, so the product of the roots should equal k.
Step 6: Calculate the product of the given roots: -3 * -4 = 12.
Step 7: Since the product of the roots equals k, we find that k = 12.

Quadratic Equations – Understanding the relationship between the coefficients of a quadratic equation and its roots, specifically using Vieta's formulas.
Roots of Equations – Calculating the sum and product of the roots to find unknown coefficients in a quadratic equation.