Solve by Completing the Square q^2-10q=20

$q^{2} - 10 q = 20$

Step 1

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

${(\frac{b}{2})}^{2} = {(- 5)}^{2}$

Step 2

Add the term to each side of the equation.

$q^{2} - 10 q + {(- 5)}^{2} = 20 + {(- 5)}^{2}$

Step 3

Simplify the equation.

Simplify the left side.

Raise $- 5$ to the power of $2$ .

$q^{2} - 10 q + 25 = 20 + {(- 5)}^{2}$

Simplify the right side.

Simplify $20 + {(- 5)}^{2}$ .

Raise $- 5$ to the power of $2$ .

$q^{2} - 10 q + 25 = 20 + 25$

Add $20$ and $25$ .

$q^{2} - 10 q + 25 = 45$

Step 4

Factor the perfect trinomial square into ${(q - 5)}^{2}$ .

${(q - 5)}^{2} = 45$

Step 5

Solve the equation for $q$ .

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

$q - 5 = \pm \sqrt{45}$

Simplify $\pm \sqrt{45}$ .

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$q - 5 = \pm \sqrt{9 (5)}$

Rewrite $9$ as $3^{2}$ .

$q - 5 = \pm \sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$q - 5 = \pm 3 \sqrt{5}$

Add $5$ to both sides of the equation.

$q = \pm 3 \sqrt{5} + 5$

Step 6

The result can be shown in multiple forms.

Exact Form:

$q = \pm 3 \sqrt{5} + 5$

Decimal Form:

$q = 11.70820393 \dots, - 1.70820393 \dots$