If the roots of the equation x^2 + 5x + k = 0 are -2 and -3, find k.

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Question: If the roots of the equation x^2 + 5x + k = 0 are -2 and -3, find k.

Options:

Correct Answer: 6

Solution:

Using Vieta\'s formulas, k = (-2)(-3) = 6.

If the roots of the equation x^2 + 5x + k = 0 are -2 and -3, find k.

If the roots of the equation x^2 + 5x + k = 0 are -2 and -3, find k.

Step 1: Understand that the equation x^2 + 5x + k = 0 is a quadratic equation.
Step 2: Recognize that the roots of the equation are given as -2 and -3.
Step 3: Recall Vieta's formulas, which tell us that for a quadratic equation ax^2 + bx + c = 0, the product of the roots (r1 and r2) is equal to c/a.
Step 4: In our equation, a = 1, b = 5, and c = k. The product of the roots -2 and -3 is (-2) * (-3).
Step 5: Calculate the product: (-2) * (-3) = 6.
Step 6: According to Vieta's formulas, this product is equal to k, so we have k = 6.

Vieta's Formulas – Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.