If the quadratic equation x^2 + 4x + k = 0 has roots -2 and -2, what is the value of k?

1 question

1 item

Q: If the quadratic equation x^2 + 4x + k = 0 has roots -2 and -2, what is the value of k?

Solution: Using the formula for roots, k = (-2)^2 - 4*(-2) = 4 + 8 = 12.

Steps: 12

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. This means both roots are the same.
Step 3: Use the formula for the roots of a quadratic equation, which is given by the expression: root = (-b ± √(b² - 4ac)) / (2a).
Step 4: In our equation, a = 1, b = 4, and c = k.
Step 5: Since both roots are -2, we can set -2 equal to the formula for the roots: -2 = (-4 ± √(4² - 4*1*k)) / (2*1).
Step 6: Simplify the equation: -2 = (-4 ± √(16 - 4k)) / 2.
Step 7: Multiply both sides by 2 to eliminate the fraction: -4 = -4 ± √(16 - 4k).
Step 8: This gives us two cases to consider: -4 = -4 + √(16 - 4k) or -4 = -4 - √(16 - 4k).
Step 9: The first case simplifies to 0 = √(16 - 4k), which means 16 - 4k = 0.
Step 10: Solve for k: 4k = 16, so k = 16 / 4 = 4.
Step 11: The second case (-4 = -4 - √(16 - 4k)) leads to a contradiction, so we ignore it.
Step 12: Therefore, the value of k is 4.