Questions: Consider the equation: x^2 - 16x + 55 = 0 A) First, use the "completing the square" process to write this equation in the form (x+D)^2 = E and enter your results below. x^2 -16 x +55 = 0 is equivalent to: (x-8)^2 0^8 = 9 >0^8 B) Solve your equation and enter your answers below as a list of numbers, separated with a comma where necessary. Answer(s):

Solution Steps

To solve the quadratic equation using the "completing the square" method, we first rearrange the equation to isolate the constant term. Then, we find the value needed to complete the square for the quadratic and linear terms. After completing the square, we solve for \(x\) by taking the square root of both sides and isolating \(x\).

To complete the square for the quadratic equation \(x^2 - 16x + 55 = 0\), we first focus on the quadratic and linear terms: \(x^2 - 16x\).

1. Take the coefficient of \(x\), which is \(-16\), divide it by 2, and square the result: \[ \left(\frac{-16}{2}\right)^2 = (-8)^2 = 64 \]

2. Add and subtract this square inside the equation: \[ x^2 - 16x + 64 - 64 + 55 = 0 \]

3. Rewrite the equation by grouping the perfect square trinomial: \[ (x - 8)^2 - 9 = 0 \]

Thus, the equation \(x^2 - 16x + 55 = 0\) is equivalent to: \[ (x - 8)^2 = 9 \]

Now, solve the equation \((x - 8)^2 = 9\).

1. Take the square root of both sides: \[ x - 8 = \pm 3 \]

2. Solve for \(x\) by considering both the positive and negative cases:

   • \(x - 8 = 3\) gives \(x = 11\)
   • \(x - 8 = -3\) gives \(x = 5\)

Final Answer