Examples

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

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

Step 1

Simplify

12 x^{2} + x (x - 3)

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

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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

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

Step 1.1.2

Multiply

x

$x$ by

x

$x$ .

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

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

Step 1.1.3

Move

- 3

$- 3$ to the left of

x

$x$ .

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

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

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

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

Step 1.2

Add

12 x^{2}

$12 x^{2}$ and

x^{2}

$x^{2}$ .

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

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

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

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

Step 2

Move all terms containing

x

$x$ to the left side of the equation.

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Step 2.1

Subtract

4 x

$4 x$ from both sides of the equation.

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

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

Step 2.2

Subtract

4 x

$4 x$ from

- 3 x

$- 3 x$ .

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

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

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

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

Step 3

Add

1

$1$ to both sides of the equation.

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

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

Step 4

Use the quadratic formula to find the solutions.

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

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

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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