Write as a Set of Linear Factors g(x)=x^2+10x+22

$g (x) = x^{2} + 10 x + 22$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = 10$ , and $c = 22$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot 22}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 22$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 22}}{2 \cdot 1}$

Multiply $- 4$ by $22$ .

$x = \frac{- 10 \pm \sqrt{100 - 88}}{2 \cdot 1}$

Subtract $88$ from $100$ .

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

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

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

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

$x = \frac{- 10 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 10 \pm 2 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 10 \pm 2 \sqrt{3}}{2}$

Simplify $\frac{- 10 \pm 2 \sqrt{3}}{2}$ .

$x = - 5 \pm \sqrt{3}$

Step 4

The final answer is the combination of both solutions.

$x = - 5 + \sqrt{3}, - 5 - \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - 5 + \sqrt{3}, - 5 - \sqrt{3}$

Decimal Form:

$x = - 3.26794919 \dots, - 6.73205080 \dots$