Final answer:
To solve the equation x^2 - 8x + 16 = 13, we find that (x - 4)^2 = 13 by recognizing the left side as a perfect square. Taking the square root of both sides gives us x = 4 ±√13, yielding two solutions: x = 4 + √13 and x = 4 - √13.
Step-by-step explanation:
To solve the quadratic equation x^2 - 8x + 16 = 13 using the square root property, let's first get the equation in the form of a perfect square on one side of the equality.
- Begin by equating the quadratic to zero: x^2 - 8x + 16 - 13 = 0
- Simplify it: x^2 - 8x + 3 = 0
- Recognize that the left side of the equation can be rewritten as a perfect square through completing the square method. Notice that (x-4)^2 gives us x^2 - 8x + 16, which is very close to our equation. Unfortunately, we have a 3 instead of 16, so this method does not work directly for this equation.
- However, if we treat this as an already completed square, we get the following: (x - 4)^2 = 13.
- Now take the square root of both sides to solve for x, remembering to consider both the positive and negative square roots: x - 4 = ±√13
- Finally, solve for x by isolating it on one side: x = 4 ±√13
Therefore, the solutions for the equation x^2 - 8x + 16 = 13 are x = 4 + √13 and x = 4 - √13.