Precalculus Examples

$6 x = 2 x^{2} + 1$

Step 1

Move all the expressions to the left side of the equation.

Tap for more steps...

Step 1.1

Subtract $2 x^{2}$ from both sides of the equation.

$6 x - 2 x^{2} = 1$

Step 1.2

Subtract $1$ from both sides of the equation.

$6 x - 2 x^{2} - 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 2$ , $b = 6$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

Multiply $- 4 \cdot - 2 \cdot - 1$ .

Tap for more steps...

Step 4.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 6 \pm \sqrt{36 + 8 \cdot - 1}}{2 \cdot - 2}$

Step 4.1.2.2

Multiply $8$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 - 8}}{2 \cdot - 2}$

Step 4.1.3

Subtract $8$ from $36$ .

$x = \frac{- 6 \pm \sqrt{28}}{2 \cdot - 2}$

Step 4.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

Tap for more steps...

Step 4.1.4.1

Factor $4$ out of $28$ .

$x = \frac{- 6 \pm \sqrt{4 (7)}}{2 \cdot - 2}$

Step 4.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot - 2}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{7}}{2 \cdot - 2}$

Step 4.2

Multiply $2$ by $- 2$ .

$x = \frac{- 6 \pm 2 \sqrt{7}}{- 4}$

Step 4.3

Simplify $\frac{- 6 \pm 2 \sqrt{7}}{- 4}$ .

$x = \frac{3 \pm \sqrt{7}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 \pm \sqrt{7}}{2}$

Decimal Form:

$x = 2.82287565 \dots, 0.17712434 \dots$