Solve (8+3x)² = (5x+3)² - (4x)²: Perfect Square Equation

To solve the equation $(8 + 3x)^2 = (5x + 3)^2 - (4x)^2$, we'll follow these steps:

• Step 1: Simplify each side of the equation using known algebraic identities.
• Step 2: Solve for $x$ after simplifying the equation.

Step 1: Let's expand each side of the equation.

The left side is $(8 + 3x)^2 = 8^2 + 2 \cdot 8 \cdot 3x + (3x)^2 = 64 + 48x + 9x^2$.

For the right side, we use the difference of squares identity:

$(5x + 3)^2 - (4x)^2 = \left((5x + 3) - (4x)\right)\left((5x + 3) + (4x)\right)$

Simplifying each:

$(5x + 3) - (4x) = x + 3$

$(5x + 3) + (4x) = 9x + 3$

So, the right side becomes $(x + 3)(9x + 3) = x(9x + 3) + 3(9x + 3) = 9x^2 + 3x + 27x + 9 = 9x^2 + 30x + 9$.

Now equate the simplified expressions from both sides:

$64 + 48x + 9x^2 = 9x^2 + 30x + 9$

Cancel the $9x^2$ terms from both sides:

$64 + 48x = 30x + 9$

Subtract $30x$ from both sides to isolate the terms:

$64 + 48x - 30x = 9$

$64 + 18x = 9$

Subtract 64 from both sides:

$18x = 9 - 64$

$18x = -55$

Divide both sides by 18 to solve for $x$:

$x = \frac{-55}{18}$

Simplifying, this gives:

$x = -3 \frac{1}{18}$

The solution to the problem is $x = -3 \frac{1}{18}$, which matches choice 3.