solve each of the following equations by completing the square and show your work
x² + 6x + 24=0
x² + 6x + 24=0
x² + 6x + 9 = -24 + 9
(x+3)^2 = -15
x+3 = ±i√15
x = -3 ±i√15
To solve the equation x² + 6x + 24 = 0 by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation:
x² + 6x = -24
Step 2: Take half of the coefficient of x (which is 6) and square it:
(6/2)² = 9
Step 3: Add the value obtained in step 2 to both sides of the equation:
x² + 6x + 9 = -24 + 9
x² + 6x + 9 = -15
Step 4: Rewrite the left side of the equation as a perfect square trinomial:
(x + 3)² = -15
Step 5: Take the square root of both sides of the equation:
√((x + 3)²) = √(-15)
Keep in mind that taking the square root of a negative number will introduce imaginary solutions.
Step 6: Simplify the equation:
x + 3 = ±√(-15)
Step 7: Subtract 3 from both sides of the equation:
x = -3 ± √(-15)
The solution to the equation x² + 6x + 24 = 0, when completed using the method of completing the square, is x = -3 ± √(-15).
Please note that when taking the square root of a negative number like -15, the solution involves imaginary numbers.