Balance and Solve: 13x² + 4x Equals 8(x+3)²

To solve the equation $13x^2 + 4x = 8(x+3)^2$, we proceed with the following steps:

• **Step 1: Expand the right-hand side.**
  We start by expanding $8(x+3)^2$:
  $(x+3)^2 = x^2 + 6x + 9$
  Multiplying by 8 gives $8(x^2 + 6x + 9) = 8x^2 + 48x + 72$.
• **Step 2: Form the standard quadratic equation.**
  Now substitute back into the initial equation:
  $13x^2 + 4x = 8x^2 + 48x + 72$
  Rearrange all terms to one side:
  $13x^2 + 4x - 8x^2 - 48x - 72 = 0$
  Simplify: $5x^2 - 44x - 72 = 0$.
• **Step 3: Apply the quadratic formula.**
  The equation is in the standard form $ax^2 + bx + c = 0$ where $a = 5$, $b = -44$, and $c = -72$.
  Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we calculate:
  $x = \frac{-(-44) \pm \sqrt{(-44)^2 - 4 \times 5 \times (-72)}}{2 \times 5}$
  $x = \frac{44 \pm \sqrt{1936 + 1440}}{10}$
  $x = \frac{44 \pm \sqrt{3376}}{10}$
  $\sqrt{3376} \approx 58.1$, so:
  $x = \frac{44 \pm 58.1}{10}$.
• **Step 4: Calculate the roots.**
  For the positive solution:
  $x_1 = \frac{44 + 58.1}{10} = 10.21$.
  For the negative solution:
  $x_2 = \frac{44 - 58.1}{10} = -1.41$.

Therefore, the solutions to the equation are $x_1 = 10.21$ and $x_2 = -1.41$. The correct choice from the given options is choice 3.