Solve Using the Square Root Property 9x^2+3x-4=0

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Trigonometry

$9 x^{2} + 3 x - 4 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 9$ , $b = 3$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (9 \cdot - 4)}}{2 \cdot 9}$

Simplify.

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Simplify the numerator.

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Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot (9 \cdot - 4)}}{2 \cdot 9}$

Multiply $9$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 36}}{2 \cdot 9}$

Multiply $- 4$ by $- 36$ .

$x = \frac{- 3 \pm \sqrt{9 + 144}}{2 \cdot 9}$

Add $9$ and $144$ .

$x = \frac{- 3 \pm \sqrt{153}}{2 \cdot 9}$

Rewrite $153$ as $3^{2} \cdot 17$ .

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Factor $9$ out of $153$ .

$x = \frac{- 3 \pm \sqrt{9 (17)}}{2 \cdot 9}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 3 \pm \sqrt{3^{2} \cdot 17}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{- 3 \pm 3 \sqrt{17}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{- 3 \pm 3 \sqrt{17}}{18}$

Simplify $\frac{- 3 \pm 3 \sqrt{17}}{18}$ .

$x = \frac{- 1 \pm \sqrt{17}}{6}$

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

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Simplify the numerator.

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Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot (9 \cdot - 4)}}{2 \cdot 9}$

Multiply $9$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 36}}{2 \cdot 9}$

Multiply $- 4$ by $- 36$ .

$x = \frac{- 3 \pm \sqrt{9 + 144}}{2 \cdot 9}$

Add $9$ and $144$ .

$x = \frac{- 3 \pm \sqrt{153}}{2 \cdot 9}$

Rewrite $153$ as $3^{2} \cdot 17$ .

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Factor $9$ out of $153$ .

$x = \frac{- 3 \pm \sqrt{9 (17)}}{2 \cdot 9}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 3 \pm \sqrt{3^{2} \cdot 17}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{- 3 \pm 3 \sqrt{17}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{- 3 \pm 3 \sqrt{17}}{18}$

Simplify $\frac{- 3 \pm 3 \sqrt{17}}{18}$ .

$x = \frac{- 1 \pm \sqrt{17}}{6}$

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{17}}{6}$

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

$x = \frac{- 1 \cdot 1 + \sqrt{17}}{6}$

Factor $- 1$ out of $\sqrt{17}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{17})}{6}$

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{17})$ .

$x = \frac{- 1 (1 - \sqrt{17})}{6}$

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{17}}{6}$

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

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Simplify the numerator.

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Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot (9 \cdot - 4)}}{2 \cdot 9}$

Multiply $9$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 36}}{2 \cdot 9}$

Multiply $- 4$ by $- 36$ .

$x = \frac{- 3 \pm \sqrt{9 + 144}}{2 \cdot 9}$

Add $9$ and $144$ .

$x = \frac{- 3 \pm \sqrt{153}}{2 \cdot 9}$

Rewrite $153$ as $3^{2} \cdot 17$ .

Tap for more steps...

Factor $9$ out of $153$ .

$x = \frac{- 3 \pm \sqrt{9 (17)}}{2 \cdot 9}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 3 \pm \sqrt{3^{2} \cdot 17}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{- 3 \pm 3 \sqrt{17}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{- 3 \pm 3 \sqrt{17}}{18}$

Simplify $\frac{- 3 \pm 3 \sqrt{17}}{18}$ .

$x = \frac{- 1 \pm \sqrt{17}}{6}$

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{17}}{6}$

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

$x = \frac{- 1 \cdot 1 - \sqrt{17}}{6}$

Factor $- 1$ out of $- \sqrt{17}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{17})}{6}$

Factor $- 1$ out of $- 1 (1) - (\sqrt{17})$ .

$x = \frac{- 1 (1 + \sqrt{17})}{6}$

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{17}}{6}$

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{17}}{6}, - \frac{1 + \sqrt{17}}{6}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{17}}{6}, - \frac{1 + \sqrt{17}}{6}$

Decimal Form:

$x = 0.52051760 \dots, - 0.85385093 \dots$

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