Solve for y y^2-12y=-34

$y^{2} - 12 y = - 34$

Step 1

Add $34$ to both sides of the equation.

$y^{2} - 12 y + 34 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = - 12$ , and $c = 34$ into the quadratic formula and solve for $y$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 34)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

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

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 34}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 34$ .

Multiply $- 4$ by $1$ .

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 34}}{2 \cdot 1}$

Multiply $- 4$ by $34$ .

$y = \frac{12 \pm \sqrt{144 - 136}}{2 \cdot 1}$

Subtract $136$ from $144$ .

$y = \frac{12 \pm \sqrt{8}}{2 \cdot 1}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

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

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

$y = 6 \pm \sqrt{2}$

Step 5

The final answer is the combination of both solutions.

$y = 6 + \sqrt{2}, 6 - \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = 6 + \sqrt{2}, 6 - \sqrt{2}$

Decimal Form:

$y = 7.41421356 \dots, 4.58578643 \dots$