Question

solve the quadratic equation by completing the square.
$x^2 + 2x - 10 = 0$
first, choose the appropriate form and fill in the blanks with the correct numbers.
then, solve the equation. round your answer to the nearest hundredth.
if there is more than one solution, separate them with commas.
form:
○ $(x + \square)^2 = \square$
○ $(x - \square)^2 = \square$
solution:
$x = \square$

Explanation:

Step1: Move the constant term

Given the quadratic equation \(x^{2}+2x - 10=0\), we first move the constant term to the right side of the equation:
\(x^{2}+2x=10\)

Step2: Complete the square

To complete the square for the expression \(x^{2}+2x\), we take half of the coefficient of \(x\) (which is \(2\)), square it, and add it to both sides. Half of \(2\) is \(1\), and squaring \(1\) gives \(1\). So we add \(1\) to both sides:
\(x^{2}+2x + 1=10 + 1\)
The left - hand side can be written as a perfect square: \((x + 1)^{2}=11\)

Step3: Solve for \(x\)

Take the square root of both sides:
\(x+1=\pm\sqrt{11}\)
Then solve for \(x\):
\(x=-1\pm\sqrt{11}\)
Calculate the numerical values: \(\sqrt{11}\approx3.32\)
So \(x=-1 + 3.32=2.32\) or \(x=-1-3.32=-4.32\)

Answer:

The form is \((x + 1)^{2}=11\), and the solutions are \(x\approx2.32,-4.32\) (or in the form of exact values \(x=-1\pm\sqrt{11}\))