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Solve for [tex]x[/tex]:

[tex]\[ -48(1 + 0.5x) + 2x^2 = 8x\left(\frac{1}{4}x + 3\right) \][/tex]


Sagot :

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]
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