Examples

$12 x^{2} + x (x - 3) = 4 x - 1$

Step 1

Simplify $12 x^{2} + x (x - 3)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$12 x^{2} + x \cdot x + x \cdot - 3 = 4 x - 1$

Step 1.1.2

Multiply $x$ by $x$ .

$12 x^{2} + x^{2} + x \cdot - 3 = 4 x - 1$

Step 1.1.3

Move $- 3$ to the left of $x$ .

$12 x^{2} + x^{2} - 3 x = 4 x - 1$

Step 1.2

Add $12 x^{2}$ and $x^{2}$ .

$13 x^{2} - 3 x = 4 x - 1$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $4 x$ from both sides of the equation.

$13 x^{2} - 3 x - 4 x = - 1$

Step 2.2

Subtract $4 x$ from $- 3 x$ .

$13 x^{2} - 7 x = - 1$

Step 3

Add $1$ to both sides of the equation.

$13 x^{2} - 7 x + 1 = 0$

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 $a = 13$ , $b = - 7$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (13 \cdot 1)}}{2 \cdot 13}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 13 \cdot 1}}{2 \cdot 13}$

Step 6.1.2

Multiply $- 4 \cdot 13 \cdot 1$ .

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Step 6.1.2.1

Multiply $- 4$ by $13$ .

$x = \frac{7 \pm \sqrt{49 - 52 \cdot 1}}{2 \cdot 13}$

Step 6.1.2.2

Multiply $- 52$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 52}}{2 \cdot 13}$

Step 6.1.3

Subtract $52$ from $49$ .

$x = \frac{7 \pm \sqrt{- 3}}{2 \cdot 13}$

Step 6.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{7 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 13}$

Step 6.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 13}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{7 \pm i \sqrt{3}}{2 \cdot 13}$

Step 6.2

Multiply $2$ by $13$ .

$x = \frac{7 \pm i \sqrt{3}}{26}$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 13 \cdot 1}}{2 \cdot 13}$

Step 7.1.2

Multiply $- 4 \cdot 13 \cdot 1$ .

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Step 7.1.2.1

Multiply $- 4$ by $13$ .

$x = \frac{7 \pm \sqrt{49 - 52 \cdot 1}}{2 \cdot 13}$

Step 7.1.2.2

Multiply $- 52$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 52}}{2 \cdot 13}$

Step 7.1.3

Subtract $52$ from $49$ .

$x = \frac{7 \pm \sqrt{- 3}}{2 \cdot 13}$

Step 7.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{7 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 13}$

Step 7.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 13}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{7 \pm i \sqrt{3}}{2 \cdot 13}$

Step 7.2

Multiply $2$ by $13$ .

$x = \frac{7 \pm i \sqrt{3}}{26}$

Step 7.3

Change the $\pm$ to $+$ .

$x = \frac{7 + i \sqrt{3}}{26}$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 13 \cdot 1}}{2 \cdot 13}$

Step 8.1.2

Multiply $- 4 \cdot 13 \cdot 1$ .

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Step 8.1.2.1

Multiply $- 4$ by $13$ .

$x = \frac{7 \pm \sqrt{49 - 52 \cdot 1}}{2 \cdot 13}$

Step 8.1.2.2

Multiply $- 52$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 52}}{2 \cdot 13}$

Step 8.1.3

Subtract $52$ from $49$ .

$x = \frac{7 \pm \sqrt{- 3}}{2 \cdot 13}$

Step 8.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{7 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 13}$

Step 8.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 13}$

Step 8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{7 \pm i \sqrt{3}}{2 \cdot 13}$

Step 8.2

Multiply $2$ by $13$ .

$x = \frac{7 \pm i \sqrt{3}}{26}$

Step 8.3

Change the $\pm$ to $-$ .

$x = \frac{7 - i \sqrt{3}}{26}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{7 + i \sqrt{3}}{26}, \frac{7 - i \sqrt{3}}{26}$

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