Solve Using the Quadratic Formula y^2+42=14y

$y^{2} + 42 = 14 y$

Step 1

Subtract $14 y$ from both sides of the equation.

$y^{2} + 42 - 14 y = 0$

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 $a = 1$ , $b = - 14$ , and $c = 42$ into the quadratic formula and solve for $y$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot 42)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 14$ to the power of $2$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot 42}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 42$ .

Multiply $- 4$ by $1$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot 42}}{2 \cdot 1}$

Multiply $- 4$ by $42$ .

$y = \frac{14 \pm \sqrt{196 - 168}}{2 \cdot 1}$

Subtract $168$ from $196$ .

$y = \frac{14 \pm \sqrt{28}}{2 \cdot 1}$

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$y = \frac{14 \pm \sqrt{4 (7)}}{2 \cdot 1}$

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

$y = \frac{14 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{14 \pm 2 \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{14 \pm 2 \sqrt{7}}{2}$

Simplify $\frac{14 \pm 2 \sqrt{7}}{2}$ .

$y = 7 \pm \sqrt{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = 7 \pm \sqrt{7}$

Decimal Form:

$y = 9.64575131 \dots, 4.35424868 \dots$