Calculus Examples

$0 = 6 x^{2} - 576 x + 7776$

Step 1

Rewrite the equation as $6 x^{2} - 576 x + 7776 = 0$ .

$6 x^{2} - 576 x + 7776 = 0$

Step 2

Factor $6$ out of $6 x^{2} - 576 x + 7776$ .

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Step 2.1

Factor $6$ out of $6 x^{2}$ .

$6 (x^{2}) - 576 x + 7776 = 0$

Step 2.2

Factor $6$ out of $- 576 x$ .

$6 (x^{2}) + 6 (- 96 x) + 7776 = 0$

Step 2.3

Factor $6$ out of $7776$ .

$6 x^{2} + 6 (- 96 x) + 6 \cdot 1296 = 0$

Step 2.4

Factor $6$ out of $6 x^{2} + 6 (- 96 x)$ .

$6 (x^{2} - 96 x) + 6 \cdot 1296 = 0$

Step 2.5

Factor $6$ out of $6 (x^{2} - 96 x) + 6 \cdot 1296$ .

$6 (x^{2} - 96 x + 1296) = 0$

Step 3

Divide each term in $6 (x^{2} - 96 x + 1296) = 0$ by $6$ and simplify.

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Step 3.1

Divide each term in $6 (x^{2} - 96 x + 1296) = 0$ by $6$ .

$\frac{6 (x^{2} - 96 x + 1296)}{6} = \frac{0}{6}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $6$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{6 (x^{2} - 96 x + 1296)}{6} = \frac{0}{6}$

Step 3.2.1.2

Divide $x^{2} - 96 x + 1296$ by $1$ .

$x^{2} - 96 x + 1296 = \frac{0}{6}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $6$ .

$x^{2} - 96 x + 1296 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = - 96$ , and $c = 1296$ into the quadratic formula and solve for $x$ .

$\frac{96 \pm \sqrt{{(- 96)}^{2} - 4 \cdot (1 \cdot 1296)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 96$ to the power of $2$ .

$x = \frac{96 \pm \sqrt{9216 - 4 \cdot 1 \cdot 1296}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 1296$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{96 \pm \sqrt{9216 - 4 \cdot 1296}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $1296$ .

$x = \frac{96 \pm \sqrt{9216 - 5184}}{2 \cdot 1}$

Step 6.1.3

Subtract $5184$ from $9216$ .

$x = \frac{96 \pm \sqrt{4032}}{2 \cdot 1}$

Step 6.1.4

Rewrite $4032$ as $24^{2} \cdot 7$ .

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Step 6.1.4.1

Factor $576$ out of $4032$ .

$x = \frac{96 \pm \sqrt{576 (7)}}{2 \cdot 1}$

Step 6.1.4.2

Rewrite $576$ as $24^{2}$ .

$x = \frac{96 \pm \sqrt{24^{2} \cdot 7}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{96 \pm 24 \sqrt{7}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{96 \pm 24 \sqrt{7}}{2}$

Step 6.3

Simplify $\frac{96 \pm 24 \sqrt{7}}{2}$ .

$x = 48 \pm 12 \sqrt{7}$

Step 7

The final answer is the combination of both solutions.

$x = 48 + 12 \sqrt{7}, 48 - 12 \sqrt{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 48 + 12 \sqrt{7}, 48 - 12 \sqrt{7}$

Decimal Form:

$x = 79.74901573 \dots, 16.25098426 \dots$