Calculus Examples

$12 x^{2} - 168 x + 405 = 0$

Step 1

Factor $3$ out of $12 x^{2} - 168 x + 405$ .

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Step 1.1

Factor $3$ out of $12 x^{2}$ .

$3 (4 x^{2}) - 168 x + 405 = 0$

Step 1.2

Factor $3$ out of $- 168 x$ .

$3 (4 x^{2}) + 3 (- 56 x) + 405 = 0$

Step 1.3

Factor $3$ out of $405$ .

$3 (4 x^{2}) + 3 (- 56 x) + 3 (135) = 0$

Step 1.4

Factor $3$ out of $3 (4 x^{2}) + 3 (- 56 x)$ .

$3 (4 x^{2} - 56 x) + 3 (135) = 0$

Step 1.5

Factor $3$ out of $3 (4 x^{2} - 56 x) + 3 (135)$ .

$3 (4 x^{2} - 56 x + 135) = 0$

Step 2

Divide each term in $3 (4 x^{2} - 56 x + 135) = 0$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 (4 x^{2} - 56 x + 135) = 0$ by $3$ .

$\frac{3 (4 x^{2} - 56 x + 135)}{3} = \frac{0}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 (4 x^{2} - 56 x + 135)}{3} = \frac{0}{3}$

Step 2.2.1.2

Divide $4 x^{2} - 56 x + 135$ by $1$ .

$4 x^{2} - 56 x + 135 = \frac{0}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $3$ .

$4 x^{2} - 56 x + 135 = 0$

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 $a = 4$ , $b = - 56$ , and $c = 135$ into the quadratic formula and solve for $x$ .

$\frac{56 \pm \sqrt{{(- 56)}^{2} - 4 \cdot (4 \cdot 135)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 56$ to the power of $2$ .

$x = \frac{56 \pm \sqrt{3136 - 4 \cdot 4 \cdot 135}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot 135$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{56 \pm \sqrt{3136 - 16 \cdot 135}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $135$ .

$x = \frac{56 \pm \sqrt{3136 - 2160}}{2 \cdot 4}$

Step 5.1.3

Subtract $2160$ from $3136$ .

$x = \frac{56 \pm \sqrt{976}}{2 \cdot 4}$

Step 5.1.4

Rewrite $976$ as $4^{2} \cdot 61$ .

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Step 5.1.4.1

Factor $16$ out of $976$ .

$x = \frac{56 \pm \sqrt{16 (61)}}{2 \cdot 4}$

Step 5.1.4.2

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

$x = \frac{56 \pm \sqrt{4^{2} \cdot 61}}{2 \cdot 4}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{56 \pm 4 \sqrt{61}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{56 \pm 4 \sqrt{61}}{8}$

Step 5.3

Simplify $\frac{56 \pm 4 \sqrt{61}}{8}$ .

$x = \frac{14 \pm \sqrt{61}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{14 + \sqrt{61}}{2}, \frac{14 - \sqrt{61}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{14 + \sqrt{61}}{2}, \frac{14 - \sqrt{61}}{2}$

Decimal Form:

$x = 10.90512483 \dots, 3.09487516 \dots$