Algebra Examples

4 x^{2} - 9 x - 9 = 0

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

Step 1

Use the quadratic formula to find the solutions.

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

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

Step 2

Substitute the values

a = 4

$a = 4$ ,

b = - 9

$b = - 9$ , and

c = - 9

$c = - 9$ into the quadratic formula and solve for

x

$x$ .

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

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise

- 9

$- 9$ to the power of

2

$2$ .

x = \frac{9 \pm \sqrt{81 - 4 \cdot 4 \cdot - 9}}{2 \cdot 4}

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 4 \cdot - 9}}{2 \cdot 4}$

Step 3.1.2

Multiply

- 4 \cdot 4 \cdot - 9

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

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Step 3.1.2.1

Multiply

- 4

$- 4$ by

4

$4$ .

x = \frac{9 \pm \sqrt{81 - 16 \cdot - 9}}{2 \cdot 4}

$x = \frac{9 \pm \sqrt{81 - 16 \cdot - 9}}{2 \cdot 4}$

Step 3.1.2.2

Multiply

- 16

$- 16$ by

- 9

$- 9$ .

x = \frac{9 \pm \sqrt{81 + 144}}{2 \cdot 4}

$x = \frac{9 \pm \sqrt{81 + 144}}{2 \cdot 4}$

x = \frac{9 \pm \sqrt{81 + 144}}{2 \cdot 4}

$x = \frac{9 \pm \sqrt{81 + 144}}{2 \cdot 4}$

Step 3.1.3

Add

81

$81$ and

144

$144$ .

x = \frac{9 \pm \sqrt{225}}{2 \cdot 4}

$x = \frac{9 \pm \sqrt{225}}{2 \cdot 4}$

Step 3.1.4

Rewrite

225

$225$ as

15^{2}

$15^{2}$ .

x = \frac{9 \pm \sqrt{15^{2}}}{2 \cdot 4}

$x = \frac{9 \pm \sqrt{15^{2}}}{2 \cdot 4}$

Step 3.1.5

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

x = \frac{9 \pm 15}{2 \cdot 4}

$x = \frac{9 \pm 15}{2 \cdot 4}$

x = \frac{9 \pm 15}{2 \cdot 4}

$x = \frac{9 \pm 15}{2 \cdot 4}$

Step 3.2

Multiply

2

$2$ by

4

$4$ .

x = \frac{9 \pm 15}{8}

$x = \frac{9 \pm 15}{8}$

x = \frac{9 \pm 15}{8}

$x = \frac{9 \pm 15}{8}$

Step 4

The final answer is the combination of both solutions.

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

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