Solve Using the Quadratic Formula 5x^2+1=4x-7

$5 x^{2} + 1 = 4 x - 7$

Step 1

Move all terms to the left side of the equation and simplify.

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

Subtract $4 x$ from both sides of the equation.

$5 x^{2} + 1 - 4 x = - 7$

Add $7$ to both sides of the equation.

$5 x^{2} + 1 - 4 x + 7 = 0$

Add $1$ and $7$ .

$5 x^{2} - 4 x + 8 = 0$

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 $a = 5$ , $b = - 4$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (5 \cdot 8)}}{2 \cdot 5}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5 \cdot 8}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 8$ .

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20 \cdot 8}}{2 \cdot 5}$

Multiply $- 20$ by $8$ .

$x = \frac{4 \pm \sqrt{16 - 160}}{2 \cdot 5}$

Subtract $160$ from $16$ .

$x = \frac{4 \pm \sqrt{- 144}}{2 \cdot 5}$

Rewrite $- 144$ as $- 1 (144)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 5}$

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 5}$

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

$x = \frac{4 \pm i \cdot \sqrt{144}}{2 \cdot 5}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 5}$

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

$x = \frac{4 \pm i \cdot 12}{2 \cdot 5}$

Move $12$ to the left of $i$ .

$x = \frac{4 \pm 12 i}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{4 \pm 12 i}{10}$

Simplify $\frac{4 \pm 12 i}{10}$ .

$x = \frac{2 \pm 6 i}{5}$