Solve Using the Square Root Property 4x^2=9-7x

$4 x^{2} = 9 - 7 x$

Step 1

Add $7 x$ to both sides of the equation.

$4 x^{2} + 7 x = 9$

Step 2

Subtract $9$ from both sides of the equation.

$4 x^{2} + 7 x - 9 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 4$ , $b = 7$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (4 \cdot - 9)}}{2 \cdot 4}$

Step 5

Simplify.

Simplify the numerator.

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 4 \cdot - 9}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 9$ .

Multiply $- 4$ by $4$ .

$x = \frac{- 7 \pm \sqrt{49 - 16 \cdot - 9}}{2 \cdot 4}$

Multiply $- 16$ by $- 9$ .

$x = \frac{- 7 \pm \sqrt{49 + 144}}{2 \cdot 4}$

Add $49$ and $144$ .

$x = \frac{- 7 \pm \sqrt{193}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 7 \pm \sqrt{193}}{8}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 4 \cdot - 9}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 9$ .

Multiply $- 4$ by $4$ .

$x = \frac{- 7 \pm \sqrt{49 - 16 \cdot - 9}}{2 \cdot 4}$

Multiply $- 16$ by $- 9$ .

$x = \frac{- 7 \pm \sqrt{49 + 144}}{2 \cdot 4}$

Add $49$ and $144$ .

$x = \frac{- 7 \pm \sqrt{193}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 7 \pm \sqrt{193}}{8}$

Change the $\pm$ to $+$ .

$x = \frac{- 7 + \sqrt{193}}{8}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + \sqrt{193}}{8}$

Factor $- 1$ out of $\sqrt{193}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{193})}{8}$

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{193})$ .

$x = \frac{- 1 (7 - \sqrt{193})}{8}$

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{193}}{8}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 4 \cdot - 9}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 9$ .

Multiply $- 4$ by $4$ .

$x = \frac{- 7 \pm \sqrt{49 - 16 \cdot - 9}}{2 \cdot 4}$

Multiply $- 16$ by $- 9$ .

$x = \frac{- 7 \pm \sqrt{49 + 144}}{2 \cdot 4}$

Add $49$ and $144$ .

$x = \frac{- 7 \pm \sqrt{193}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 7 \pm \sqrt{193}}{8}$

Change the $\pm$ to $-$ .

$x = \frac{- 7 - \sqrt{193}}{8}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - \sqrt{193}}{8}$

Factor $- 1$ out of $- \sqrt{193}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{193})}{8}$

Factor $- 1$ out of $- 1 (7) - (\sqrt{193})$ .

$x = \frac{- 1 (7 + \sqrt{193})}{8}$

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{193}}{8}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{193}}{8}, - \frac{7 + \sqrt{193}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7 - \sqrt{193}}{8}, - \frac{7 + \sqrt{193}}{8}$

Decimal Form:

$x = 0.86155549 \dots, - 2.61155549 \dots$

Solve Using the Square Root Property 4x^2=9-7x

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