Find All Complex Solutions 4x^2-7x=3x+24

$4 x^{2} - 7 x = 3 x + 24$

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.

$4 x^{2} - 7 x - 3 x = 24$

Subtract $24$ from both sides of the equation.

$4 x^{2} - 7 x - 3 x - 24 = 0$

Subtract $3 x$ from $- 7 x$ .

$4 x^{2} - 10 x - 24 = 0$

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 $a = 4$ , $b = - 10$ , and $c = - 24$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (4 \cdot - 24)}}{2 \cdot 4}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 4 \cdot - 24}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 24$ .

Multiply $- 4$ by $4$ .

$x = \frac{10 \pm \sqrt{100 - 16 \cdot - 24}}{2 \cdot 4}$

Multiply $- 16$ by $- 24$ .

$x = \frac{10 \pm \sqrt{100 + 384}}{2 \cdot 4}$

Add $100$ and $384$ .

$x = \frac{10 \pm \sqrt{484}}{2 \cdot 4}$

Rewrite $484$ as $22^{2}$ .

$x = \frac{10 \pm \sqrt{22^{2}}}{2 \cdot 4}$

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

$x = \frac{10 \pm 22}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{10 \pm 22}{8}$

Simplify $\frac{10 \pm 22}{8}$ .

$x = \frac{5 \pm 11}{4}$

Step 5

The final answer is the combination of both solutions.

$x = 4, - \frac{3}{2}$