To solve the equation by completing the square, we're given the quadratic equation [tex]\( x^2 + 2x = 17 \)[/tex]. Let's solve it step by step:
1. Move the constant term to the other side of the equation:
[tex]\[
x^2 + 2x - 17 = 0
\][/tex]
2. Identify the coefficient of [tex]\( x \)[/tex] and the leading coefficient:
In this equation, the coefficient of [tex]\( x \)[/tex] is 2 and the leading coefficient (coefficient of [tex]\( x^2 \)[/tex]) is 1.
3. Rewrite the equation to isolate the quadratic and linear terms on one side:
[tex]\[
x^2 + 2x = 17
\][/tex]
4. Complete the square:
To complete the square, take half of the coefficient of [tex]\( x \)[/tex] and square it. The coefficient of [tex]\( x \)[/tex] is 2, so:
[tex]\[
\left(\frac{2}{2}\right)^2 = 1
\][/tex]
Add and subtract this value inside the equation to complete the square:
[tex]\[
x^2 + 2x + 1 - 1 = 17
\][/tex]
[tex]\[
x^2 + 2x + 1 = 17 + 1
\][/tex]
[tex]\[
(x + 1)^2 = 18
\][/tex]
5. Take the square root of both sides:
[tex]\[
x + 1 = \pm \sqrt{18}
\][/tex]
Simplify [tex]\( \sqrt{18} \)[/tex]:
[tex]\[
\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
\][/tex]
Thus, we have:
[tex]\[
x + 1 = \pm 3\sqrt{2}
\][/tex]
6. Isolate [tex]\( x \)[/tex] by subtracting 1 from both sides:
[tex]\[
x = -1 \pm 3\sqrt{2}
\][/tex]
7. Calculate the two solutions:
[tex]\[
x = -1 + 3\sqrt{2} \quad \text{and} \quad x = -1 - 3\sqrt{2}
\][/tex]
8. Approximate the values to the nearest hundredth:
[tex]\[
3\sqrt{2} \approx 4.24
\][/tex]
Therefore:
[tex]\[
x \approx -1 + 4.24 = 3.24
\][/tex]
[tex]\[
x \approx -1 - 4.24 = -5.24
\][/tex]
9. The solutions are:
[tex]\[
\boxed{3.24, -5.24}
\][/tex]
So, after completing the square and solving, the solutions to the equation [tex]\( x^2 + 2x = 17 \)[/tex], rounded to the nearest hundredth, are [tex]\( 3.24 \)[/tex] and [tex]\( -5.24 \)[/tex].
