Solve by Completing the Square x^2-18x+77=0

$x^{2} - 18 x + 77 = 0$

Step 1

Subtract $77$ from both sides of the equation.

$x^{2} - 18 x = - 77$

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- 9)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 18 x + {(- 9)}^{2} = - 77 + {(- 9)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $- 9$ to the power of $2$ .

$x^{2} - 18 x + 81 = - 77 + {(- 9)}^{2}$

Simplify the right side.

Simplify $- 77 + {(- 9)}^{2}$ .

Raise $- 9$ to the power of $2$ .

$x^{2} - 18 x + 81 = - 77 + 81$

Add $- 77$ and $81$ .

$x^{2} - 18 x + 81 = 4$

Step 5

Factor the perfect trinomial square into ${(x - 9)}^{2}$ .

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

Step 6

Solve the equation for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 9 = \pm \sqrt{4}$

Simplify $\pm \sqrt{4}$ .

Rewrite $4$ as $2^{2}$ .

$x - 9 = \pm \sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x - 9 = \pm 2$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x - 9 = 2$

Move all terms not containing $x$ to the right side of the equation.

Add $9$ to both sides of the equation.

$x = 2 + 9$

Add $2$ and $9$ .

$x = 11$

Next, use the negative value of the $\pm$ to find the second solution.

$x - 9 = - 2$

Move all terms not containing $x$ to the right side of the equation.

Add $9$ to both sides of the equation.

$x = - 2 + 9$

Add $- 2$ and $9$ .

$x = 7$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 11, 7$

Solve by Completing the Square x^2-18x+77=0

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