Let's solve the equation [tex]\(9(2x - 3)^2 + 7 = 448\)[/tex] step-by-step:
1. Isolate the quadratic term:
We start by subtracting 7 from both sides of the equation:
[tex]\[
9(2x - 3)^2 + 7 - 7 = 448 - 7
\][/tex]
This simplifies to:
[tex]\[
9(2x - 3)^2 = 441
\][/tex]
2. Divide by 9:
Next, we divide both sides of the equation by 9 to further isolate the squared term:
[tex]\[
\frac{9(2x - 3)^2}{9} = \frac{441}{9}
\][/tex]
Simplifying this, we get:
[tex]\[
(2x - 3)^2 = 49
\][/tex]
3. Take the square root of both sides:
To solve for the expression inside the square, we take the square root of both sides. Remember to consider both the positive and negative roots:
[tex]\[
2x - 3 = \pm \sqrt{49}
\][/tex]
Since [tex]\(\sqrt{49} = 7\)[/tex], we have:
[tex]\[
2x - 3 = 7 \quad \text{or} \quad 2x - 3 = -7
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Now we solve each of these equations separately:
For [tex]\(2x - 3 = 7\)[/tex]:
[tex]\[
2x - 3 = 7
\][/tex]
Add 3 to both sides:
[tex]\[
2x = 10
\][/tex]
Divide by 2:
[tex]\[
x = 5
\][/tex]
For [tex]\(2x - 3 = -7\)[/tex]:
[tex]\[
2x - 3 = -7
\][/tex]
Add 3 to both sides:
[tex]\[
2x = -4
\][/tex]
Divide by 2:
[tex]\[
x = -2
\][/tex]
Thus, the solutions to the equation [tex]\(9(2x - 3)^2 + 7 = 448\)[/tex] are [tex]\(x = 5\)[/tex] and [tex]\(x = -2\)[/tex].
So the correct answer is:
B) [tex]\(x = 5, -2\)[/tex]
