Solve for g g^2-10g+3=0

$g^{2} - 10 g + 3 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 10$ , and $c = 3$ into the quadratic formula and solve for $g$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$g = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$g = \frac{10 \pm \sqrt{100 - 4 \cdot 3}}{2 \cdot 1}$

Multiply $- 4$ by $3$ .

$g = \frac{10 \pm \sqrt{100 - 12}}{2 \cdot 1}$

Subtract $12$ from $100$ .

$g = \frac{10 \pm \sqrt{88}}{2 \cdot 1}$

Rewrite $88$ as $2^{2} \cdot 22$ .

Factor $4$ out of $88$ .

$g = \frac{10 \pm \sqrt{4 (22)}}{2 \cdot 1}$

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

$g = \frac{10 \pm \sqrt{2^{2} \cdot 22}}{2 \cdot 1}$

Pull terms out from under the radical.

$g = \frac{10 \pm 2 \sqrt{22}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$g = \frac{10 \pm 2 \sqrt{22}}{2}$

Simplify $\frac{10 \pm 2 \sqrt{22}}{2}$ .

$g = 5 \pm \sqrt{22}$

The final answer is the combination of both solutions.

$g = 5 + \sqrt{22}, 5 - \sqrt{22}$

The result can be shown in multiple forms.

Exact Form:

$g = 5 + \sqrt{22}, 5 - \sqrt{22}$

Decimal Form:

$g = 9.69041575 \dots, 0.30958424 \dots$