Question

solve by completing the square.
$x^2 + 2x - 51 = -6$
select the correct response:
$-1, -11$
$5, -9$
$8 + \sqrt{106}, 8 - \sqrt{106}$
$-1 + \sqrt{46}, -1 - \sqrt{46}$

Explanation:

Step1: Move constants to right

First, we add 51 and 6 to both sides of the equation \(x^{2}+2x - 51=-6\) to get the quadratic terms and linear term on the left and constants on the right.
\(x^{2}+2x=- 6 + 51\)
\(x^{2}+2x = 45\)

Step2: Complete the square

To complete the square for the expression \(x^{2}+2x\), we take half of the coefficient of \(x\) (which is \(2\)), square it. Half of \(2\) is \(1\), and squaring \(1\) gives \(1\). We add this to both sides of the equation:
\(x^{2}+2x + 1=45 + 1\)
The left side is now a perfect square trinomial, which factors as \((x + 1)^{2}\), so we have:
\((x + 1)^{2}=46\)

Step3: Solve for x

Take the square root of both sides:
\(x+1=\pm\sqrt{46}\)
Then subtract \(1\) from both sides to solve for \(x\):
\(x=-1\pm\sqrt{46}\)
Which gives the solutions \(x=-1 + \sqrt{46}\) and \(x=-1-\sqrt{46}\)

Answer:

\(-1 + \sqrt{46}\), \(-1-\sqrt{46}\) (corresponding to the option \(-1 + \sqrt{46}\), \(-1 - \sqrt{46}\))