Solve for y y^2-10y+29=0

$y^{2} - 10 y + 29 = 0$

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 $a = 1$ , $b = - 10$ , and $c = 29$ into the quadratic formula and solve for $y$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 29)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 29$ .

Multiply $- 4$ by $1$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 29}}{2 \cdot 1}$

Multiply $- 4$ by $29$ .

$y = \frac{10 \pm \sqrt{100 - 116}}{2 \cdot 1}$

Subtract $116$ from $100$ .

$y = \frac{10 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$y = \frac{10 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$y = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{10 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$y = \frac{10 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

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

$y = \frac{10 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$y = \frac{10 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{10 \pm 4 i}{2}$

Simplify $\frac{10 \pm 4 i}{2}$ .

$y = 5 \pm 2 i$

Step 4

The final answer is the combination of both solutions.

$y = 5 + 2 i, 5 - 2 i$