Final answer:
To solve the quadratic equation by completing the square, we transform it into a perfect square trinomial, take the square root of both sides, and then solve for x, yielding two solutions. The correct final answer is option d: x = 8 ± 6√2.
Step-by-step explanation:
We are tasked with solving the quadratic equation x²−16x−8=0 using the method of completing the square. Completing the square involves rewriting our equation in the form (x-h)²=k, where h and k are numbers we will determine.
First step is to move the constant term to the other side of the equation: x²−16x = 8.
Now, we add the square of half the coefficient of x to both sides. Half of −16 is −8, and the square of −8 is 64. So we add 64 to both sides: x²−16x + 64 = 72. This makes the left side a perfect square.
Now we can write the equation as (x−8)² = 72. Taking the square root of both sides gives us x−8 = ±√72. Adding 8 to both sides gives us the solutions x = 8 ± √72.
We simplify √72 to 6√2 to get x = 8 ± 6√2.
Therefore, the values of x that satisfy the equation x²−16x−8=0 are x = 8 + 6√2 and x = 8 - 6√2, which aligns with option d.