Algebra Examples

$x^{2} = \frac{2}{3} (x + 1)$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the right side.

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Step 1.1.1

Simplify $\frac{2}{3} \cdot (x + 1)$ .

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Step 1.1.1.1

Apply the distributive property.

$x^{2} = \frac{2}{3} x + \frac{2}{3} \cdot 1$

Step 1.1.1.2

Combine $\frac{2}{3}$ and $x$ .

$x^{2} = \frac{2 x}{3} + \frac{2}{3} \cdot 1$

Step 1.1.1.3

Multiply $\frac{2}{3}$ by $1$ .

$x^{2} = \frac{2 x}{3} + \frac{2}{3}$

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Subtract $\frac{2 x}{3}$ from both sides of the equation.

$x^{2} - \frac{2 x}{3} = \frac{2}{3}$

Step 1.2.2

Subtract $\frac{2}{3}$ from both sides of the equation.

$x^{2} - \frac{2 x}{3} - \frac{2}{3} = 0$

Step 2

Multiply through by the least common denominator $3$ , then simplify.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

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

Step 2.2.1.2

Cancel the common factor.

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

Step 2.2.1.3

Rewrite the expression.

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

Step 2.2.2

Cancel the common factor of $3$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

$3 x^{2} - 2 x - 2 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 3$ , $b = - 2$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 2}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 2$ .

$x = \frac{2 \pm \sqrt{4 + 24}}{2 \cdot 3}$

Step 5.1.3

Add $4$ and $24$ .

$x = \frac{2 \pm \sqrt{28}}{2 \cdot 3}$

Step 5.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 5.1.4.1

Factor $4$ out of $28$ .

$x = \frac{2 \pm \sqrt{4 (7)}}{2 \cdot 3}$

Step 5.1.4.2

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

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

Step 5.1.5

Pull terms out from under the radical.

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

Step 5.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{7}}{6}$

Step 5.3

Simplify $\frac{2 \pm 2 \sqrt{7}}{6}$ .

$x = \frac{1 \pm \sqrt{7}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 \pm \sqrt{7}}{3}$

Decimal Form:

$x = 1.21525043 \dots, - 0.54858377 \dots$