Pre-Algebra Examples

$3 x^{2} = 7 (x - 3)$

Step 1

Simplify $7 (x - 3)$ .

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Step 1.1

Apply the distributive property.

$3 x^{2} = 7 x + 7 \cdot - 3$

Step 1.2

Multiply $7$ by $- 3$ .

$3 x^{2} = 7 x - 21$

Step 2

Subtract $7 x$ from both sides of the equation.

$3 x^{2} - 7 x = - 21$

Step 3

Add $21$ to both sides of the equation.

$3 x^{2} - 7 x + 21 = 0$

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 $a = 3$ , $b = - 7$ , and $c = 21$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (3 \cdot 21)}}{2 \cdot 3}$

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 $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 3 \cdot 21}}{2 \cdot 3}$

Step 6.1.2

Multiply $- 4 \cdot 3 \cdot 21$ .

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Step 6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{7 \pm \sqrt{49 - 12 \cdot 21}}{2 \cdot 3}$

Step 6.1.2.2

Multiply $- 12$ by $21$ .

$x = \frac{7 \pm \sqrt{49 - 252}}{2 \cdot 3}$

Step 6.1.3

Subtract $252$ from $49$ .

$x = \frac{7 \pm \sqrt{- 203}}{2 \cdot 3}$

Step 6.1.4

Rewrite $- 203$ as $- 1 (203)$ .

$x = \frac{7 \pm \sqrt{- 1 \cdot 203}}{2 \cdot 3}$

Step 6.1.5

Rewrite $\sqrt{- 1 (203)}$ as $\sqrt{- 1} \cdot \sqrt{203}$ .

$x = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{203}}{2 \cdot 3}$

Step 6.1.6

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

$x = \frac{7 \pm i \sqrt{203}}{2 \cdot 3}$

Step 6.2

Multiply $2$ by $3$ .

$x = \frac{7 \pm i \sqrt{203}}{6}$

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 $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 3 \cdot 21}}{2 \cdot 3}$

Step 7.1.2

Multiply $- 4 \cdot 3 \cdot 21$ .

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Step 7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{7 \pm \sqrt{49 - 12 \cdot 21}}{2 \cdot 3}$

Step 7.1.2.2

Multiply $- 12$ by $21$ .

$x = \frac{7 \pm \sqrt{49 - 252}}{2 \cdot 3}$

Step 7.1.3

Subtract $252$ from $49$ .

$x = \frac{7 \pm \sqrt{- 203}}{2 \cdot 3}$

Step 7.1.4

Rewrite $- 203$ as $- 1 (203)$ .

$x = \frac{7 \pm \sqrt{- 1 \cdot 203}}{2 \cdot 3}$

Step 7.1.5

Rewrite $\sqrt{- 1 (203)}$ as $\sqrt{- 1} \cdot \sqrt{203}$ .

$x = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{203}}{2 \cdot 3}$

Step 7.1.6

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

$x = \frac{7 \pm i \sqrt{203}}{2 \cdot 3}$

Step 7.2

Multiply $2$ by $3$ .

$x = \frac{7 \pm i \sqrt{203}}{6}$

Step 7.3

Change the $\pm$ to $+$ .

$x = \frac{7 + i \sqrt{203}}{6}$

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 $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 3 \cdot 21}}{2 \cdot 3}$

Step 8.1.2

Multiply $- 4 \cdot 3 \cdot 21$ .

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Step 8.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{7 \pm \sqrt{49 - 12 \cdot 21}}{2 \cdot 3}$

Step 8.1.2.2

Multiply $- 12$ by $21$ .

$x = \frac{7 \pm \sqrt{49 - 252}}{2 \cdot 3}$

Step 8.1.3

Subtract $252$ from $49$ .

$x = \frac{7 \pm \sqrt{- 203}}{2 \cdot 3}$

Step 8.1.4

Rewrite $- 203$ as $- 1 (203)$ .

$x = \frac{7 \pm \sqrt{- 1 \cdot 203}}{2 \cdot 3}$

Step 8.1.5

Rewrite $\sqrt{- 1 (203)}$ as $\sqrt{- 1} \cdot \sqrt{203}$ .

$x = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{203}}{2 \cdot 3}$

Step 8.1.6

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

$x = \frac{7 \pm i \sqrt{203}}{2 \cdot 3}$

Step 8.2

Multiply $2$ by $3$ .

$x = \frac{7 \pm i \sqrt{203}}{6}$

Step 8.3

Change the $\pm$ to $-$ .

$x = \frac{7 - i \sqrt{203}}{6}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{7 + i \sqrt{203}}{6}, \frac{7 - i \sqrt{203}}{6}$