Finite Math Examples

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

Step 1

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

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

Step 2

Simplify $\frac{\sqrt{x}}{2} - \frac{x^{2}}{\sqrt{x}}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $\frac{x^{2}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{x}}{2} - (\frac{x^{2}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}) = 0$

Step 2.1.2

Combine and simplify the denominator.

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Step 2.1.2.1

Multiply $\frac{x^{2}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 2.1.2.2

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

$\frac{\sqrt{x}}{2} - \frac{x^{2} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}} = 0$

Step 2.1.2.3

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

$\frac{\sqrt{x}}{2} - \frac{x^{2} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}} = 0$

Step 2.1.2.4

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

$\frac{\sqrt{x}}{2} - \frac{x^{2} \sqrt{x}}{{\sqrt{x}}^{1 + 1}} = 0$

Step 2.1.2.5

Add $1$ and $1$ .

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

Step 2.1.2.6

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

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Step 2.1.2.6.1

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

$\frac{\sqrt{x}}{2} - \frac{x^{2} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}} = 0$

Step 2.1.2.6.2

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

$\frac{\sqrt{x}}{2} - \frac{x^{2} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}} = 0$

Step 2.1.2.6.3

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

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

Step 2.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.1.2.6.4.1

Cancel the common factor.

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

Step 2.1.2.6.4.2

Rewrite the expression.

$\frac{\sqrt{x}}{2} - \frac{x^{2} \sqrt{x}}{x^{1}} = 0$

Step 2.1.2.6.5

Simplify.

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

Step 2.1.3

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.1.3.1

Factor $x$ out of $x^{2} \sqrt{x}$ .

$\frac{\sqrt{x}}{2} - \frac{x (x \sqrt{x})}{x} = 0$

Step 2.1.3.2

Cancel the common factors.

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Step 2.1.3.2.1

Raise $x$ to the power of $1$ .

$\frac{\sqrt{x}}{2} - \frac{x (x \sqrt{x})}{x^{1}} = 0$

Step 2.1.3.2.2

Factor $x$ out of $x^{1}$ .

$\frac{\sqrt{x}}{2} - \frac{x (x \sqrt{x})}{x \cdot 1} = 0$

Step 2.1.3.2.3

Cancel the common factor.

$\frac{\sqrt{x}}{2} - \frac{x (x \sqrt{x})}{x \cdot 1} = 0$

Step 2.1.3.2.4

Rewrite the expression.

$\frac{\sqrt{x}}{2} - \frac{x \sqrt{x}}{1} = 0$

Step 2.1.3.2.5

Divide $x \sqrt{x}$ by $1$ .

$\frac{\sqrt{x}}{2} - (x \sqrt{x}) = 0$

$\frac{\sqrt{x}}{2} - x \sqrt{x} = 0$

Step 2.2

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

$\frac{\sqrt{x}}{2} - x \sqrt{x} \cdot \frac{2}{2} = 0$

Step 2.3

Combine $- x \sqrt{x}$ and $\frac{2}{2}$ .

$\frac{\sqrt{x}}{2} + \frac{- x \sqrt{x} \cdot 2}{2} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{\sqrt{x} - x \sqrt{x} \cdot 2}{2} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $\sqrt{x}$ out of $\sqrt{x} - x \sqrt{x} \cdot 2$ .

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Step 2.5.1.1

Multiply by $1$ .

$\frac{\sqrt{x} \cdot 1 - x \sqrt{x} \cdot 2}{2} = 0$

Step 2.5.1.2

Factor $\sqrt{x}$ out of $- x \sqrt{x} \cdot 2$ .

$\frac{\sqrt{x} \cdot 1 + \sqrt{x} (- x \cdot 2)}{2} = 0$

Step 2.5.1.3

Factor $\sqrt{x}$ out of $\sqrt{x} \cdot 1 + \sqrt{x} (- x \cdot 2)$ .

$\frac{\sqrt{x} (1 - x \cdot 2)}{2} = 0$

Step 2.5.2

Multiply $2$ by $- 1$ .

$\frac{\sqrt{x} (1 - 2 x)}{2} = 0$

Step 3

Set the numerator equal to zero.

$\sqrt{x} (1 - 2 x) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sqrt{x} = 0$

$1 - 2 x = 0$

Step 4.2

Set $\sqrt{x}$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $\sqrt{x}$ equal to $0$ .

$\sqrt{x} = 0$

Step 4.2.2

Solve $\sqrt{x} = 0$ for $x$ .

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Step 4.2.2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = 0^{2}$

Step 4.2.2.2

Simplify each side of the equation.

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Step 4.2.2.2.1

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

${(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2.2.2

Simplify the left side.

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Step 4.2.2.2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 4.2.2.2.2.1.2

Simplify.

$x = 0^{2}$

Step 4.2.2.2.3

Simplify the right side.

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Step 4.2.2.2.3.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 4.3

Set $1 - 2 x$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $1 - 2 x$ equal to $0$ .

$1 - 2 x = 0$

Step 4.3.2

Solve $1 - 2 x = 0$ for $x$ .

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Step 4.3.2.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 4.3.2.2

Divide each term in $- 2 x = - 1$ by $- 2$ and simplify.

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Step 4.3.2.2.1

Divide each term in $- 2 x = - 1$ by $- 2$ .

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

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.3.2.2.2.1.1

Cancel the common factor.

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

Step 4.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.3.2.2.3

Simplify the right side.

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Step 4.3.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 4.4

The final solution is all the values that make $\sqrt{x} (1 - 2 x) = 0$ true.

$x = 0, \frac{1}{2}$