Basic Math Examples

$4 C^{2} - 6 = - 12 C$

Step 1

Add $12 C$ to both sides of the equation.

$4 C^{2} - 6 + 12 C = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $2$ out of $4 C^{2} - 6 + 12 C$ .

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Step 2.1.1

Factor $2$ out of $4 C^{2}$ .

$2 (2 C^{2}) - 6 + 12 C = 0$

Step 2.1.2

Factor $2$ out of $- 6$ .

$2 (2 C^{2}) + 2 (- 3) + 12 C = 0$

Step 2.1.3

Factor $2$ out of $12 C$ .

$2 (2 C^{2}) + 2 (- 3) + 2 (6 C) = 0$

Step 2.1.4

Factor $2$ out of $2 (2 C^{2}) + 2 (- 3)$ .

$2 (2 C^{2} - 3) + 2 (6 C) = 0$

Step 2.1.5

Factor $2$ out of $2 (2 C^{2} - 3) + 2 (6 C)$ .

$2 (2 C^{2} - 3 + 6 C) = 0$

Step 2.2

Reorder terms.

$2 (2 C^{2} + 6 C - 3) = 0$

Step 3

Divide each term in $2 (2 C^{2} + 6 C - 3) = 0$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 (2 C^{2} + 6 C - 3) = 0$ by $2$ .

$\frac{2 (2 C^{2} + 6 C - 3)}{2} = \frac{0}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 (2 C^{2} + 6 C - 3)}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $2 C^{2} + 6 C - 3$ by $1$ .

$2 C^{2} + 6 C - 3 = \frac{0}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $2$ .

$2 C^{2} + 6 C - 3 = 0$

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 $a = 2$ , $b = 6$ , and $c = - 3$ into the quadratic formula and solve for $C$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (2 \cdot - 3)}}{2 \cdot 2}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $6$ to the power of $2$ .

$C = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2 \cdot - 3}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 3$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$C = \frac{- 6 \pm \sqrt{36 - 8 \cdot - 3}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 3$ .

$C = \frac{- 6 \pm \sqrt{36 + 24}}{2 \cdot 2}$

Step 6.1.3

Add $36$ and $24$ .

$C = \frac{- 6 \pm \sqrt{60}}{2 \cdot 2}$

Step 6.1.4

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 6.1.4.1

Factor $4$ out of $60$ .

$C = \frac{- 6 \pm \sqrt{4 (15)}}{2 \cdot 2}$

Step 6.1.4.2

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

$C = \frac{- 6 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 2}$

Step 6.1.5

Pull terms out from under the radical.

$C = \frac{- 6 \pm 2 \sqrt{15}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$C = \frac{- 6 \pm 2 \sqrt{15}}{4}$

Step 6.3

Simplify $\frac{- 6 \pm 2 \sqrt{15}}{4}$ .

$C = \frac{- 3 \pm \sqrt{15}}{2}$

Step 7

The final answer is the combination of both solutions.

$C = - \frac{3 - \sqrt{15}}{2}, - \frac{3 + \sqrt{15}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$C = - \frac{3 - \sqrt{15}}{2}, - \frac{3 + \sqrt{15}}{2}$

Decimal Form:

$C = 0.43649167 \dots, - 3.43649167 \dots$