Consider the equation: x^2+12x+27=0 Eand A) First, use the "completing the square" process to write this equation in the form (x+D)^2=E enter your results below. x^2+12x+27=0 is equivalent to: square =square Preview left side of eqn: B) Solve your equation and enter your answers below as a list of numbers, separated with a comma where necessary. Answer(s): square

To solve the equation \(x^2 + 12x + 27 = 0\) by completing the square, follow these steps:<br /><br />### Step 1: Move the constant term to the other side of the equation<br />\[ x^2 + 12x + 27 = 0 \]<br />\[ x^2 + 12x = -27 \]<br /><br />### Step 2: Complete the square<br />To complete the square, we need to add and subtract the square of half the coefficient of \(x\). The coefficient of \(x\) is 12, so half of it is 6, and its square is \(6^2 = 36\).<br /><br />Add and subtract 36 on the left side:<br />\[ x^2 + 12x + 36 - 36 = -27 \]<br />\[ x^2 + 12x + 36 = -27 + 36 \]<br />\[ (x + 6)^2 = 9 \]<br /><br />So, the equation in the form \((x + D)^2 = E\) is:<br />\[ (x + 6)^2 = 9 \]<br /><br />### Step 3: Solve the equation<br />Take the square root of both sides:<br />\[ x + 6 = \pm 3 \]<br /><br />Solve for \(x\):<br />\[ x + 6 = 3 \quad \text{or} \quad x + 6 = -3 \]<br />\[ x = 3 - 6 \quad \text{or} \quad x = -3 - 6 \]<br />\[ x = -3 \quad \text{or} \quad x = -9 \]<br /><br />### Final Answers<br />A) The equation \(x^2 + 12x + 27 = 0\) is equivalent to:<br />\[ (x + 6)^2 = 9 \]<br /><br />B) The solutions to the equation are:<br />\[ -3, -9 \]