Solve Using the Quadratic Formula 5x-7x^2=3x+4

$5 x - 7 x^{2} = 3 x + 4$

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 $3 x$ from both sides of the equation.

$5 x - 7 x^{2} - 3 x = 4$

Subtract $4$ from both sides of the equation.

$5 x - 7 x^{2} - 3 x - 4 = 0$

Subtract $3 x$ from $5 x$ .

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

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 $a = - 7$ , $b = 2$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (- 7 \cdot - 4)}}{2 \cdot - 7}$

Step 4

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot - 7 \cdot - 4$ .

Multiply $- 4$ by $- 7$ .

$x = \frac{- 2 \pm \sqrt{4 + 28 \cdot - 4}}{2 \cdot - 7}$

Multiply $28$ by $- 4$ .

$x = \frac{- 2 \pm \sqrt{4 - 112}}{2 \cdot - 7}$

Subtract $112$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 108}}{2 \cdot - 7}$

Rewrite $- 108$ as $- 1 (108)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 108}}{2 \cdot - 7}$

Rewrite $\sqrt{- 1 (108)}$ as $\sqrt{- 1} \cdot \sqrt{108}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{108}}{2 \cdot - 7}$

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

$x = \frac{- 2 \pm i \cdot \sqrt{108}}{2 \cdot - 7}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$x = \frac{- 2 \pm i \cdot \sqrt{36 (3)}}{2 \cdot - 7}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{6^{2} \cdot 3}}{2 \cdot - 7}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (6 \sqrt{3})}{2 \cdot - 7}$

Move $6$ to the left of $i$ .

$x = \frac{- 2 \pm 6 i \sqrt{3}}{2 \cdot - 7}$

Multiply $2$ by $- 7$ .

$x = \frac{- 2 \pm 6 i \sqrt{3}}{- 14}$

Simplify $\frac{- 2 \pm 6 i \sqrt{3}}{- 14}$ .

$x = \frac{1 \pm 3 i \sqrt{3}}{7}$

Solve Using the Quadratic Formula 5x-7x^2=3x+4

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