Solve Quadratic Equation: 4x² + 24x + 36 = 0 Step-by-Step

To solve the quadratic equation $4x^2 + 24x + 36 = 0$, we will simplify it by factoring:

First, notice that the given equation can be simplified as a perfect square:

• Recognize that $4x^2$, $24x$, and $36$ can form a perfect square trinomial: $(2x + 6)^2$.
• Expand $(2x + 6)^2$ to verify it corresponds to the original equation:
  $(2x + 6)^2 = (2x + 6)(2x + 6) = 4x^2 + 12x + 12x + 36 = 4x^2 + 24x + 36$.

The equation has now been verified to be a perfect square: $\left(2x + 6\right)^2 = 0$.

Set $2x + 6 = 0$, and solve for $x$:

• Subtract 6 from both sides: $2x = -6$.
• Divide both sides by 2: $x = -3$.

Thus, the solution to the quadratic equation is $\boxed{x = -3}$.