Calculus Examples

$12 x^{2} - 264 x + 989 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 12$ , $b = - 264$ , and $c = 989$ into the quadratic formula and solve for $x$ .

$\frac{264 \pm \sqrt{{(- 264)}^{2} - 4 \cdot (12 \cdot 989)}}{2 \cdot 12}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 264$ to the power of $2$ .

$x = \frac{264 \pm \sqrt{69696 - 4 \cdot 12 \cdot 989}}{2 \cdot 12}$

Step 3.1.2

Multiply $- 4 \cdot 12 \cdot 989$ .

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Step 3.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{264 \pm \sqrt{69696 - 48 \cdot 989}}{2 \cdot 12}$

Step 3.1.2.2

Multiply $- 48$ by $989$ .

$x = \frac{264 \pm \sqrt{69696 - 47472}}{2 \cdot 12}$

Step 3.1.3

Subtract $47472$ from $69696$ .

$x = \frac{264 \pm \sqrt{22224}}{2 \cdot 12}$

Step 3.1.4

Rewrite $22224$ as $4^{2} \cdot 1389$ .

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Step 3.1.4.1

Factor $16$ out of $22224$ .

$x = \frac{264 \pm \sqrt{16 (1389)}}{2 \cdot 12}$

Step 3.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{264 \pm \sqrt{4^{2} \cdot 1389}}{2 \cdot 12}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{264 \pm 4 \sqrt{1389}}{2 \cdot 12}$

Step 3.2

Multiply $2$ by $12$ .

$x = \frac{264 \pm 4 \sqrt{1389}}{24}$

Step 3.3

Simplify $\frac{264 \pm 4 \sqrt{1389}}{24}$ .

$x = \frac{66 \pm \sqrt{1389}}{6}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{66 + \sqrt{1389}}{6}, \frac{66 - \sqrt{1389}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{66 + \sqrt{1389}}{6}, \frac{66 - \sqrt{1389}}{6}$

Decimal Form:

$x = 17.21154838 \dots, 4.78845161 \dots$