Sure, let's solve the equation [tex]\( x^2 + 3x = 25 \)[/tex] by completing the square, and then we'll determine the answer from the given options.
1. Start with the given equation:
[tex]\[
x^2 + 3x = 25
\][/tex]
2. Move the constant term to the right side of the equation to set it up for completing the square:
[tex]\[
x^2 + 3x - 25 = 0
\][/tex]
3. To complete the square, we need to add and subtract the square of half the coefficient of [tex]\( x \)[/tex]. The coefficient of [tex]\( x \)[/tex] is [tex]\( 3 \)[/tex], so half of it is [tex]\( 1.5 \)[/tex], and its square is [tex]\( (1.5)^2 = 2.25 \)[/tex].
4. Add and subtract [tex]\( 2.25 \)[/tex] to the left side of the equation:
[tex]\[
x^2 + 3x + 2.25 - 2.25 = 25
\][/tex]
[tex]\[
x^2 + 3x + 2.25 = 25 + 2.25
\][/tex]
[tex]\[
x^2 + 3x + 2.25 = 27.25
\][/tex]
5. Now, recognize that the left side is a perfect square trinomial:
[tex]\[
(x + 1.5)^2 = 27.25
\][/tex]
6. Take the square root of both sides of the equation:
[tex]\[
x + 1.5 = \pm \sqrt{27.25}
\][/tex]
7. Simplify the square root. We can approximate [tex]\( \sqrt{27.25} \approx 5.22 \)[/tex].
8. Thus, we have two possible equations:
[tex]\[
x + 1.5 = 5.22 \quad \text{or} \quad x + 1.5 = -5.22
\][/tex]
9. Solve for [tex]\( x \)[/tex] in each case:
[tex]\[
x = 5.22 - 1.5 \quad \text{or} \quad x = -5.22 - 1.5
\][/tex]
[tex]\[
x \approx 3.72 \quad \text{or} \quad x \approx -6.72
\][/tex]
10. Therefore, the solutions to the equation [tex]\( x^2 + 3x = 25 \)[/tex] are approximately [tex]\( 3.72 \)[/tex] and [tex]\( -6.72 \)[/tex], rounded to the nearest hundredth.
From the given options, the correct answer is:
[tex]\( 3.72, -6.72 \)[/tex].
