Solve by Completing the Square x^2-2x-31=-7

$x^{2} - 2 x - 31 = - 7$

Step 1

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

Add $31$ to both sides of the equation.

$x^{2} - 2 x = - 7 + 31$

Add $- 7$ and $31$ .

$x^{2} - 2 x = 24$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 1)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 2 x + {(- 1)}^{2} = 24 + {(- 1)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $- 1$ to the power of $2$ .

$x^{2} - 2 x + 1 = 24 + {(- 1)}^{2}$

Simplify the right side.

Simplify $24 + {(- 1)}^{2}$ .

Raise $- 1$ to the power of $2$ .

$x^{2} - 2 x + 1 = 24 + 1$

Add $24$ and $1$ .

$x^{2} - 2 x + 1 = 25$

Step 5

Factor the perfect trinomial square into ${(x - 1)}^{2}$ .

${(x - 1)}^{2} = 25$

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.

$x - 1 = \pm \sqrt{25}$

Simplify $\pm \sqrt{25}$ .

Rewrite $25$ as $5^{2}$ .

$x - 1 = \pm \sqrt{5^{2}}$

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

$x - 1 = \pm 5$

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.

$x - 1 = 5$

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

Add $1$ to both sides of the equation.

$x = 5 + 1$

Add $5$ and $1$ .

$x = 6$

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

$x - 1 = - 5$

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

Add $1$ to both sides of the equation.

$x = - 5 + 1$

Add $- 5$ and $1$ .

$x = - 4$

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

$x = 6, - 4$

Solve by Completing the Square x^2-2x-31=-7

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