Answer:
- Form: (x -5)² -2
- Solution: x = 5+√2, 5-√2
Explanation:
You want to solve the quadratic x² -10x +23 = 0 by completing the square.
Form
The constant term in a perfect square trinomial is the square of half the coefficient of the linear term. Here, the linear term is -10x, so the constant will need to be (-10/2)² = 25. It is 23, so we can add and subtract 2 to get the form we need:
x² -10x +23 +2 -2 = 0
x² -10x +25 -2 = 0
(x -5)² -2 = 0 . . . . . . . . . . the equation in vertex form
Solution
We can add 2 and take the square root to find the solutions.
(x -5)² = 2 . . . . . . . . add 2 to both sides
x -5 = ±√2 . . . . . . . square root
x = 5 ± √2 . . . . . . add 5
The two solutions are ...
5 +√2, 5 -√2
__
Additional comments
If you don't want to think too hard about how to get the "k" in the vertex form (x -h)² +k, you can simply add and subtract the constant you found you need (25). This looks like ...
x² -10x +25 +23 -25 = 0
(x -5)² -2 = 0 . . . . as above
Sometimes this is easier than thinking, "I need 25 and I have 23, so I need to add 2 (and subtract 2)."
We add and subtract the same number so we don't change equation. Equivalently, we can add the same number on both sides of the equation:
x² -10x +23 +2 = 2 . . . . . 2 added to both sides
(x -5)² = 2 . . . . . . . . . . . a correct equation, but not the expression this problem is looking for in the "Form" section.
You will note the sign (minus) in the binomial square (x-5)² is the same as the sign of the x-coefficient (-10x).
<95141404393>