Solve Using the Quadratic Formula y^2-12y+31=0

$y^{2} - 12 y + 31 = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = 1$ , $b = - 12$ , and $c = 31$ into the quadratic formula and solve for $y$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 31)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 31}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 31$ .

Multiply $- 4$ by $1$ .

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 31}}{2 \cdot 1}$

Multiply $- 4$ by $31$ .

$y = \frac{12 \pm \sqrt{144 - 124}}{2 \cdot 1}$

Subtract $124$ from $144$ .

$y = \frac{12 \pm \sqrt{20}}{2 \cdot 1}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$y = \frac{12 \pm \sqrt{4 (5)}}{2 \cdot 1}$

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

$y = \frac{12 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{12 \pm 2 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{12 \pm 2 \sqrt{5}}{2}$

Simplify $\frac{12 \pm 2 \sqrt{5}}{2}$ .

$y = 6 \pm \sqrt{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = 6 \pm \sqrt{5}$

Decimal Form:

$y = 8.23606797 \dots, 3.76393202 \dots$