Sure, let's break down the process of solving the given equation step by step:
The equation we have is:
[tex]\[ -48(1 + 0.5x) + 2x^2 = 8x \left( \frac{1}{4}x + 3 \right) \][/tex]
### Step 1: Simplify Both Sides
First, expand and simplify both sides of the equation.
#### Left-Hand Side (LHS):
[tex]\[ -48(1 + 0.5x) + 2x^2 \][/tex]
Distribute the [tex]\(-48\)[/tex]:
[tex]\[ -48 \cdot 1 + (-48) \cdot (0.5x) + 2x^2 \][/tex]
[tex]\[ -48 - 24x + 2x^2 \][/tex]
So, the LHS becomes:
[tex]\[ 2x^2 - 24x - 48 \][/tex]
#### Right-Hand Side (RHS):
[tex]\[ 8x \left( \frac{1}{4}x + 3 \right) \][/tex]
Distribute the [tex]\(8x\)[/tex]:
[tex]\[ 8x \cdot \frac{1}{4}x + 8x \cdot 3 \][/tex]
[tex]\[ 2x^2 + 24x \][/tex]
So, the RHS becomes:
[tex]\[ 2x^2 + 24x \][/tex]
### Step 2: Set the Equations Equal
Now set the simplified LHS equal to the simplified RHS:
[tex]\[ 2x^2 - 24x - 48 = 2x^2 + 24x \][/tex]
### Step 3: Move All Terms to One Side
Subtract [tex]\(2x^2 + 24x\)[/tex] from both sides:
[tex]\[ 2x^2 - 24x - 48 - 2x^2 - 24x = 0 \][/tex]
[tex]\[ -48x - 48 = 0 \][/tex]
### Step 4: Simplify Further
Isolate [tex]\(x\)[/tex]:
[tex]\[ -48x - 48 = 0 \][/tex]
Add [tex]\(48\)[/tex] to both sides:
[tex]\[ -48x = 48 \][/tex]
Divide by [tex]\(-48\)[/tex]:
[tex]\[ x = \frac{48}{-48} \][/tex]
[tex]\[ x = -1 \][/tex]
### Conclusion
The solution to the equation is:
[tex]\[ x = -1 \][/tex]
So, [tex]\(x = -1\)[/tex] is the value that satisfies the given equation:
[tex]\[ -48(1+0.5 x) + 2 x^2 = 8 x \left(\frac{1}{4} x + 3\right) \][/tex]
The final answer is:
[tex]\[ x = -1 \][/tex]
