Question

Solve the quadratic equation by completing the square. x^2-9x-22=0Completing the square gives us: (x- Answer )^2 = AnswerEnter your solutions below from smallest to largest. If a solution is repeated type that answer for both values of x. If your answer is not an integer then type it as a decimal rounded to the nearest hundredth.x=Answer and x=Answer

Answer 1

Answer::

$\begin{gathered} (x-\frac{9}{2})^2=\frac{169}{4} \\ x=-2\text{ and x=11} \end{gathered}$

Explanation:

Given the quadratic equation:

$x^2-9x-22=0$

To solve it by completing the square, follow the steps below:

Step 1: Take the constant to the right-hand side.

$x^2-9x=22$

Step 2: Divide the coefficient of x by 2, square it and add it to both sides.

$x^2-9x+(-\frac{9}{2})^2=22+(-\frac{9}{2})^2$

Step 3: Write the left-hand side as a perfect square.

$(x-\frac{9}{2})^2=\frac{169}{4}$

Step 4: Take the square root of both sides.

$\begin{gathered} \sqrt{(x-\frac{9}{2})^2}=\pm\sqrt{\frac{169}{4}} \\ x-\frac{9}{2}=\pm\frac{13}{2} \end{gathered}$

Step 5: Solve for x.

$\begin{gathered} x=\frac{9}{2}\pm\frac{13}{2}=\frac{9\pm13}{2} \\ \implies x=\frac{9+13}{2}\text{ or }x=\frac{9-13}{2}\text{ } \\ x=11\text{ or }x=-2 \end{gathered}$

So, we have:

$\begin{gathered} Completing\; the\; square\; gives\; us\colon(x-\frac{9}{2})^2=\frac{169}{4} \\ x=-2\text{ and x=11} \end{gathered}$