Algebra Examples

$2 (5 - 5 x^{2}) = 5$

Step 1

Divide each term in $2 (5 - 5 x^{2}) = 5$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 (5 - 5 x^{2}) = 5$ by $2$ .

$\frac{2 (5 - 5 x^{2})}{2} = \frac{5}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 (5 - 5 x^{2})}{2} = \frac{5}{2}$

Step 1.2.1.2

Divide $5 - 5 x^{2}$ by $1$ .

$5 - 5 x^{2} = \frac{5}{2}$

Step 2

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

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Step 2.1

Subtract $5$ from both sides of the equation.

$- 5 x^{2} = \frac{5}{2} - 5$

Step 2.2

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

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

Step 2.3

Combine $- 5$ and $\frac{2}{2}$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 5$ by $2$ .

$- 5 x^{2} = \frac{5 - 10}{2}$

Step 2.5.2

Subtract $10$ from $5$ .

$- 5 x^{2} = \frac{- 5}{2}$

Step 2.6

Move the negative in front of the fraction.

$- 5 x^{2} = - \frac{5}{2}$

Step 3

Divide each term in $- 5 x^{2} = - \frac{5}{2}$ by $- 5$ and simplify.

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Step 3.1

Divide each term in $- 5 x^{2} = - \frac{5}{2}$ by $- 5$ .

$\frac{- 5 x^{2}}{- 5} = \frac{- \frac{5}{2}}{- 5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 5 x^{2}}{- 5} = \frac{- \frac{5}{2}}{- 5}$

Step 3.2.1.2

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

$x^{2} = \frac{- \frac{5}{2}}{- 5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3.2

Cancel the common factor of $5$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

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

Step 3.3.2.2

Factor $5$ out of $- 5$ .

$x^{2} = \frac{5 (- 1)}{2} \cdot \frac{1}{- 5}$

Step 3.3.2.3

Factor $5$ out of $- 5$ .

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

Step 3.3.2.4

Cancel the common factor.

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

Step 3.3.2.5

Rewrite the expression.

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

Step 3.3.3

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

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

Step 3.3.4

Multiply $2$ by $- 1$ .

$x^{2} = \frac{- 1}{- 2}$

Step 3.3.5

Dividing two negative values results in a positive value.

$x^{2} = \frac{1}{2}$

Step 4

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

$x = \pm \sqrt{\frac{1}{2}}$

Step 5

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{2}}$

Step 5.3

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

$x = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.4

Combine and simplify the denominator.

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Step 5.4.1

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

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

Step 5.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 5.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 5.4.4

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

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.4.6

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

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Step 5.4.6.1

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

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

Step 5.4.6.2

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

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

Step 5.4.6.3

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

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.4.6.4

Cancel the common factor of $2$ .

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Step 5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2}}{2^{1}}$

Step 5.4.6.5

Evaluate the exponent.

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

Step 6

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

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Step 6.1

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

$x = \frac{\sqrt{2}}{2}$

Step 6.2

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

$x = - \frac{\sqrt{2}}{2}$

Step 6.3

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

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Decimal Form:

$x = 0.70710678 \dots, - 0.70710678 \dots$