Solve Using the Quadratic Formula x^2-12x+21=0

$x^{2} - 12 x + 21 = 0$

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 $a = 1$ , $b = - 12$ , and $c = 21$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 21)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 21}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 21$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 21}}{2 \cdot 1}$

Multiply $- 4$ by $21$ .

$x = \frac{12 \pm \sqrt{144 - 84}}{2 \cdot 1}$

Subtract $84$ from $144$ .

$x = \frac{12 \pm \sqrt{60}}{2 \cdot 1}$

Rewrite $60$ as $2^{2} \cdot 15$ .

Factor $4$ out of $60$ .

$x = \frac{12 \pm \sqrt{4 (15)}}{2 \cdot 1}$

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

$x = \frac{12 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{15}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{15}}{2}$

Simplify $\frac{12 \pm 2 \sqrt{15}}{2}$ .

$x = 6 \pm \sqrt{15}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 6 \pm \sqrt{15}$

Decimal Form:

$x = 9.87298334 \dots, 2.12701665 \dots$