Solve Using the Quadratic Formula 3x^2-7x-3=x

$3 x^{2} - 7 x - 3 = x$

Step 1

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

Subtract $x$ from both sides of the equation.

$3 x^{2} - 7 x - 3 - x = 0$

Subtract $x$ from $- 7 x$ .

$3 x^{2} - 8 x - 3 = 0$

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 $a = 3$ , $b = - 8$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (3 \cdot - 3)}}{2 \cdot 3}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 3}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 3$ .

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot - 3}}{2 \cdot 3}$

Multiply $- 12$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 + 36}}{2 \cdot 3}$

Add $64$ and $36$ .

$x = \frac{8 \pm \sqrt{100}}{2 \cdot 3}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{8 \pm \sqrt{10^{2}}}{2 \cdot 3}$

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

$x = \frac{8 \pm 10}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{8 \pm 10}{6}$

Simplify $\frac{8 \pm 10}{6}$ .

$x = \frac{4 \pm 5}{3}$

Step 5

The final answer is the combination of both solutions.

$x = 3, - \frac{1}{3}$