Algebra Examples

$f (x) = \frac{5}{2} \sqrt{- (x - 6)}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{5}{2} \cdot \sqrt{- (x - 6)}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{5}{2} \cdot \sqrt{- (x - 6)} = 0$ .

$\frac{5}{2} \cdot \sqrt{- (x - 6)} = 0$

Step 1.2.2

Multiply both sides of the equation by $\frac{2}{5}$ .

$\frac{2}{5} (\frac{5}{2} \cdot \sqrt{- (x - 6)}) = \frac{2}{5} \cdot 0$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $\frac{2}{5} (\frac{5}{2} \cdot \sqrt{- (x - 6)})$ .

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Step 1.2.3.1.1.1

Combine $\frac{5}{2}$ and $\sqrt{- (x - 6)}$ .

$\frac{2}{5} \cdot \frac{5 \sqrt{- (x - 6)}}{2} = \frac{2}{5} \cdot 0$

Step 1.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.2.1

Cancel the common factor.

$\frac{2}{5} \cdot \frac{5 \sqrt{- (x - 6)}}{2} = \frac{2}{5} \cdot 0$

Step 1.2.3.1.1.2.2

Rewrite the expression.

$\frac{1}{5} (5 \sqrt{- (x - 6)}) = \frac{2}{5} \cdot 0$

Step 1.2.3.1.1.3

Cancel the common factor of $5$ .

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Step 1.2.3.1.1.3.1

Factor $5$ out of $5 \sqrt{- (x - 6)}$ .

$\frac{1}{5} (5 (\sqrt{- (x - 6)})) = \frac{2}{5} \cdot 0$

Step 1.2.3.1.1.3.2

Cancel the common factor.

$\frac{1}{5} (5 \sqrt{- (x - 6)}) = \frac{2}{5} \cdot 0$

Step 1.2.3.1.1.3.3

Rewrite the expression.

$\sqrt{- (x - 6)} = \frac{2}{5} \cdot 0$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $\frac{2}{5}$ by $0$ .

$\sqrt{- (x - 6)} = 0$

Step 1.2.4

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

${\sqrt{- (x - 6)}}^{2} = 0^{2}$

Step 1.2.5

Simplify each side of the equation.

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Step 1.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- (x - 6)}$ as ${(- (x - 6))}^{\frac{1}{2}}$ .

${({(- (x - 6))}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Simplify ${({(- (x - 6))}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.5.2.1.1

Multiply the exponents in ${({(- (x - 6))}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.5.2.1.1.1

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

${(- (x - 6))}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.5.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.5.2.1.1.2.1

Cancel the common factor.

${(- (x - 6))}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.5.2.1.1.2.2

Rewrite the expression.

${(- (x - 6))}^{1} = 0^{2}$

Step 1.2.5.2.1.2

Apply the distributive property.

${(- x - - 6)}^{1} = 0^{2}$

Step 1.2.5.2.1.3

Multiply.

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Step 1.2.5.2.1.3.1

Multiply $- 1$ by $- 6$ .

${(- x + 6)}^{1} = 0^{2}$

Step 1.2.5.2.1.3.2

Simplify.

$- x + 6 = 0^{2}$

Step 1.2.5.3

Simplify the right side.

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Step 1.2.5.3.1

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

$- x + 6 = 0$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Subtract $6$ from both sides of the equation.

$- x = - 6$

Step 1.2.6.2

Divide each term in $- x = - 6$ by $- 1$ and simplify.

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Step 1.2.6.2.1

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Step 1.2.6.2.2

Simplify the left side.

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Step 1.2.6.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Step 1.2.6.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Step 1.2.6.2.3

Simplify the right side.

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Step 1.2.6.2.3.1

Divide $- 6$ by $- 1$ .

$x = 6$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(6, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{5}{2} \cdot \sqrt{- ((0) - 6)}$

Step 2.2

Simplify $\frac{5}{2} \cdot \sqrt{- ((0) - 6)}$ .

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Subtract $6$ from $0$ .

$y = \frac{5}{2} \cdot \sqrt{- - 6}$

Step 2.2.1.2

Multiply $- 1$ by $- 6$ .

$y = \frac{5}{2} \cdot \sqrt{6}$

Step 2.2.2

Combine $\frac{5}{2}$ and $\sqrt{6}$ .

$y = \frac{5 \sqrt{6}}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{5 \sqrt{6}}{2})$

Step 3

List the intersections.

x-intercept(s): $(6, 0)$

y-intercept(s): $(0, \frac{5 \sqrt{6}}{2})$

Step 4