If the roots of the equation x^2 + 4x + k = 0 are -2 and -2, what is the value o

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Question: If the roots of the equation x^2 + 4x + k = 0 are -2 and -2, what is the value of k? (2023)

Options:

Correct Answer: 4

Exam Year: 2023

Solution:

Using the formula for the sum of roots, -2 + -2 = -4, and product of roots, (-2)(-2) = 4, we find k = 4.

If the roots of the equation x^2 + 4x + k = 0 are -2 and -2, what is the value of k? (2023)

If the roots of the equation x^2 + 4x + k = 0 are -2 and -2, what is the value of k? (2023)

Step 1: Identify the given quadratic equation, which is x^2 + 4x + k = 0.
Step 2: Note that the roots of the equation are given as -2 and -2.
Step 3: Use the formula for the sum of the roots, which states that the sum of the roots (r1 + r2) is equal to -b/a. Here, b = 4 and a = 1.
Step 4: Calculate the sum of the roots: -2 + -2 = -4.
Step 5: Set the sum of the roots equal to -b/a: -4 = -4 (this confirms the roots are correct).
Step 6: Use the formula for the product of the roots, which states that the product of the roots (r1 * r2) is equal to c/a. Here, c = k and a = 1.
Step 7: Calculate the product of the roots: (-2) * (-2) = 4.
Step 8: Set the product of the roots equal to c/a: 4 = k (since a = 1).
Step 9: Conclude that the value of k is 4.

Quadratic Equations – Understanding the properties of roots of quadratic equations, including the sum and product of roots.
Discriminant – Recognizing that a quadratic equation can have real and repeated roots based on the discriminant.