Question

solve the quadratic equation by completing the square.
$x^{2}+2x - 9=0$
first, choose the appropriate form and fill in the blanks with the correct numbers. then, solve the equation. simplify your answer as much as possible. if there is more than one solution, separate them with commas.
form:
$\circ(x + \square)^{2}=\square$
$\circ(x - \square)^{2}=\square$
solution:
$x = \square$

Explanation:

Step1: Rewrite the equation for completing the square

Given $x^{2}+2x - 9=0$, we focus on the $x^{2}+2x$ terms. For the quadratic expression $x^{2}+bx$, to complete the square we add $(\frac{b}{2})^{2}$. Here $b = 2$, so $(\frac{b}{2})^{2}=(\frac{2}{2})^{2}=1$. Add 1 to both sides of the equation: $x^{2}+2x+1-9 = 1$, which can be rewritten as $(x + 1)^{2}-9=1$.

Step2: Isolate the squared - term

Add 9 to both sides of the equation $(x + 1)^{2}-9=1$. We get $(x + 1)^{2}=1 + 9$, so $(x + 1)^{2}=10$.

Step3: Solve for x

Take the square - root of both sides: $x+1=\pm\sqrt{10}$. Then subtract 1 from both sides: $x=-1\pm\sqrt{10}$.

Answer:

Form: $(x + 1)^{2}=10$
Solution: $x=-1+\sqrt{10},-1 - \sqrt{10}$