Write as a Set of Linear Factors f(x)=x^2-6x+25

$f (x) = x^{2} - 6 x + 25$

Step 1

Use the quadratic formula to find the roots for $x^{2} - 6 x + 25 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 6$ , and $c = 25$ into the quadratic formula and solve for $x$ .

$f (x) = \frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 25$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 25}}{2 \cdot 1}$

Multiply $- 4$ by $25$ .

$x = \frac{6 \pm \sqrt{36 - 100}}{2 \cdot 1}$

Subtract $100$ from $36$ .

$x = \frac{6 \pm \sqrt{- 64}}{2 \cdot 1}$

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot 8}{2 \cdot 1}$

Move $8$ to the left of $i$ .

$x = \frac{6 \pm 8 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 8 i}{2}$

Simplify $\frac{6 \pm 8 i}{2}$ .

$x = 3 \pm 4 i$

Step 2

Find the factors from the roots, then multiply the factors together.

$f (x) = (x - (3 + 4 i)) (x - (3 - 4 i))$