To solve the equation [tex]\(-48(1 + 0.5x) + 2x^2 = 8x \left(\frac{1}{4}x + 3 \right)\)[/tex], we need to follow these steps:
1. Expand both sides of the equation:
- Expand the left-hand side: [tex]\(-48(1 + 0.5x) + 2x^2\)[/tex]
[tex]\[
-48 \cdot 1 - 48 \cdot 0.5x + 2x^2 = -48 - 24x + 2x^2
\][/tex]
- Expand the right-hand side: [tex]\(8x \left(\frac{1}{4}x + 3\right)\)[/tex]
[tex]\[
8x \cdot \frac{1}{4}x + 8x \cdot 3 = 2x^2 + 24x
\][/tex]
So the expanded equation is:
[tex]\[
-48 - 24x + 2x^2 = 2x^2 + 24x
\][/tex]
2. Move all terms to one side to set the equation to 0:
Subtract [tex]\(2x^2\)[/tex] and [tex]\(24x\)[/tex] from both sides of the equation:
[tex]\[
-48 - 24x + 2x^2 - 2x^2 - 24x = 2x^2 + 24x - 2x^2 - 24x
\][/tex]
This simplifies to:
[tex]\[
-48 - 48x = 0
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add [tex]\(48\)[/tex] to both sides of the equation:
[tex]\[
-48x = 48
\][/tex]
Divide both sides by [tex]\(-48\)[/tex]:
[tex]\[
x = -1
\][/tex]
So, the solution to the equation [tex]\(-48(1 + 0.5x) + 2x^2 = 8x \left(\frac{1}{4}x + 3 \right)\)[/tex] is:
[tex]\[
x = -1
\][/tex]
