Question

Which perfect square equation represents x²+10x+23=0 by completing the square?

a. (x+5)²=-3
b. (x+5)²=2
c. (x+10)²=2
d. (x+5)²=-2

Answer 1

To complete the square for the quadratic equation x² + 10x + 23 = 0, we add and subtract 25 to form a perfect square trinomial, resulting in the expression (x + 5)² - 2 = 0. After rearranging, we find that the correct completed square equation is (x + 5)² = -2, which matches option d.

Step-by-step explanation:

The student is asking about completing the square for the quadratic equation x² + 10x + 23 = 0. To complete the square, we look for a perfect square trinomial on the left side of the equation. This involves taking the coefficient of x, which is 10, dividing by 2 to get 5, and then squaring it to obtain 25. Thus, we need to add and subtract 25 from the left side of the equation to maintain equality.

Here are the steps to transform the given equation:

1. Start with the original equation: x² + 10x + 23 = 0.
2. Add and subtract 25 after the x term: x² + 10x + 25 - 25 + 23 = 0.
3. Combine like terms and structure it as a perfect square: (x + 5)² - 2 = 0.
4. Add 2 to both sides: (x + 5)² = 2.

The perfect square equation that represents the original equation by completing the square is therefore (x + 5)² = -2, which corresponds to option d.