Pre-Algebra Examples

$\frac{4}{7} x^{2} + \frac{3}{5} = \frac{2}{7}$

Step 1

Combine $\frac{4}{7}$ and $x^{2}$ .

$\frac{4 x^{2}}{7} + \frac{3}{5} = \frac{2}{7}$

Step 2

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 2.1

Subtract $\frac{3}{5}$ from both sides of the equation.

$\frac{4 x^{2}}{7} = \frac{2}{7} - \frac{3}{5}$

Step 2.2

To write $\frac{2}{7}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4 x^{2}}{7} = \frac{2}{7} \cdot \frac{5}{5} - \frac{3}{5}$

Step 2.3

To write $- \frac{3}{5}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{4 x^{2}}{7} = \frac{2}{7} \cdot \frac{5}{5} - \frac{3}{5} \cdot \frac{7}{7}$

Step 2.4

Write each expression with a common denominator of $35$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.4.1

Multiply $\frac{2}{7}$ by $\frac{5}{5}$ .

$\frac{4 x^{2}}{7} = \frac{2 \cdot 5}{7 \cdot 5} - \frac{3}{5} \cdot \frac{7}{7}$

Step 2.4.2

Multiply $7$ by $5$ .

$\frac{4 x^{2}}{7} = \frac{2 \cdot 5}{35} - \frac{3}{5} \cdot \frac{7}{7}$

Step 2.4.3

Multiply $\frac{3}{5}$ by $\frac{7}{7}$ .

$\frac{4 x^{2}}{7} = \frac{2 \cdot 5}{35} - \frac{3 \cdot 7}{5 \cdot 7}$

Step 2.4.4

Multiply $5$ by $7$ .

$\frac{4 x^{2}}{7} = \frac{2 \cdot 5}{35} - \frac{3 \cdot 7}{35}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{4 x^{2}}{7} = \frac{2 \cdot 5 - 3 \cdot 7}{35}$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Multiply $2$ by $5$ .

$\frac{4 x^{2}}{7} = \frac{10 - 3 \cdot 7}{35}$

Step 2.6.2

Multiply $- 3$ by $7$ .

$\frac{4 x^{2}}{7} = \frac{10 - 21}{35}$

Step 2.6.3

Subtract $21$ from $10$ .

$\frac{4 x^{2}}{7} = \frac{- 11}{35}$

Step 2.7

Move the negative in front of the fraction.

$\frac{4 x^{2}}{7} = - \frac{11}{35}$

Step 3

Multiply both sides of the equation by $\frac{7}{4}$ .

$\frac{7}{4} \cdot \frac{4 x^{2}}{7} = \frac{7}{4} (- \frac{11}{35})$

Step 4

Simplify both sides of the equation.

Tap for more steps...

Step 4.1

Simplify the left side.

Tap for more steps...

Step 4.1.1

Simplify $\frac{7}{4} \cdot \frac{4 x^{2}}{7}$ .

Tap for more steps...

Step 4.1.1.1

Combine.

$\frac{7 (4 x^{2})}{4 \cdot 7} = \frac{7}{4} (- \frac{11}{35})$

Step 4.1.1.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.1.1.2.1

Cancel the common factor.

$\frac{7 (4 x^{2})}{4 \cdot 7} = \frac{7}{4} (- \frac{11}{35})$

Step 4.1.1.2.2

Rewrite the expression.

$\frac{4 x^{2}}{4} = \frac{7}{4} (- \frac{11}{35})$

Step 4.1.1.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.1.1.3.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{7}{4} (- \frac{11}{35})$

Step 4.1.1.3.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{7}{4} (- \frac{11}{35})$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $\frac{7}{4} (- \frac{11}{35})$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.2.1.1.1

Move the leading negative in $- \frac{11}{35}$ into the numerator.

$x^{2} = \frac{7}{4} \cdot \frac{- 11}{35}$

Step 4.2.1.1.2

Factor $7$ out of $35$ .

$x^{2} = \frac{7}{4} \cdot \frac{- 11}{7 (5)}$

Step 4.2.1.1.3

Cancel the common factor.

$x^{2} = \frac{7}{4} \cdot \frac{- 11}{7 \cdot 5}$

Step 4.2.1.1.4

Rewrite the expression.

$x^{2} = \frac{1}{4} \cdot \frac{- 11}{5}$

Step 4.2.1.2

Multiply $\frac{1}{4}$ by $\frac{- 11}{5}$ .

$x^{2} = \frac{- 11}{4 \cdot 5}$

Step 4.2.1.3

Simplify the expression.

Tap for more steps...

Step 4.2.1.3.1

Multiply $4$ by $5$ .

$x^{2} = \frac{- 11}{20}$

Step 4.2.1.3.2

Move the negative in front of the fraction.

$x^{2} = - \frac{11}{20}$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{11}{20}}$

Step 6

Simplify $\pm \sqrt{- \frac{11}{20}}$ .

Tap for more steps...

Step 6.1

Rewrite $- \frac{11}{20}$ as ${(\frac{i}{2})}^{2} \frac{11}{5}$ .

Tap for more steps...

Step 6.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{11}{20}}$

Step 6.1.2

Factor the perfect power $1^{2}$ out of $11$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 11}{20}}$

Step 6.1.3

Factor the perfect power $2^{2}$ out of $20$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 11}{2^{2} \cdot 5}}$

Step 6.1.4

Rearrange the fraction $\frac{1^{2} \cdot 11}{2^{2} \cdot 5}$ .

$x = \pm \sqrt{i^{2} ({(\frac{1}{2})}^{2} \frac{11}{5})}$

Step 6.1.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

$x = \pm \sqrt{{(\frac{i}{2})}^{2} \frac{11}{5}}$

Step 6.2

Pull terms out from under the radical.

$x = \pm \frac{i}{2} \sqrt{\frac{11}{5}}$

Step 6.3

Rewrite $\sqrt{\frac{11}{5}}$ as $\frac{\sqrt{11}}{\sqrt{5}}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11}}{\sqrt{5}}$

Step 6.4

Multiply $\frac{\sqrt{11}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{i}{2} (\frac{\sqrt{11}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 6.5

Combine and simplify the denominator.

Tap for more steps...

Step 6.5.1

Multiply $\frac{\sqrt{11}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 6.5.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 6.5.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 6.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 6.5.5

Add $1$ and $1$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 6.5.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Step 6.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 6.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 6.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 6.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.5.6.4.1

Cancel the common factor.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 6.5.6.4.2

Rewrite the expression.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{5^{1}}$

Step 6.5.6.5

Evaluate the exponent.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11} \sqrt{5}}{5}$

Step 6.6

Simplify the numerator.

Tap for more steps...

Step 6.6.1

Combine using the product rule for radicals.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{11 \cdot 5}}{5}$

Step 6.6.2

Multiply $11$ by $5$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{55}}{5}$

Step 6.7

Multiply $\frac{i}{2} \cdot \frac{\sqrt{55}}{5}$ .

Tap for more steps...

Step 6.7.1

Multiply $\frac{i}{2}$ by $\frac{\sqrt{55}}{5}$ .

$x = \pm \frac{i \sqrt{55}}{2 \cdot 5}$

Step 6.7.2

Multiply $2$ by $5$ .

$x = \pm \frac{i \sqrt{55}}{10}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{i \sqrt{55}}{10}$

Step 7.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{i \sqrt{55}}{10}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{i \sqrt{55}}{10}, - \frac{i \sqrt{55}}{10}$