Algebra Examples

$f (x) = - 4 \sqrt{x + 5} + 6$

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 $x$ .

$0 = - 4 \sqrt{x + 5} + 6$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 4 \sqrt{x + 5} + 6 = 0$ .

$- 4 \sqrt{x + 5} + 6 = 0$

Step 1.2.2

Subtract $6$ from both sides of the equation.

$- 4 \sqrt{x + 5} = - 6$

Step 1.2.3

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

${(- 4 \sqrt{x + 5})}^{2} = {(- 6)}^{2}$

Step 1.2.4

Simplify each side of the equation.

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Step 1.2.4.1

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

${(- 4 {(x + 5)}^{\frac{1}{2}})}^{2} = {(- 6)}^{2}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Simplify ${(- 4 {(x + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.1.1

Apply the product rule to $- 4 {(x + 5)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(x + 5)}^{\frac{1}{2}})}^{2} = {(- 6)}^{2}$

Step 1.2.4.2.1.2

Raise $- 4$ to the power of $2$ .

$16 {({(x + 5)}^{\frac{1}{2}})}^{2} = {(- 6)}^{2}$

Step 1.2.4.2.1.3

Multiply the exponents in ${({(x + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.1.3.1

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

$16 {(x + 5)}^{\frac{1}{2} \cdot 2} = {(- 6)}^{2}$

Step 1.2.4.2.1.3.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.3.2.1

Cancel the common factor.

$16 {(x + 5)}^{\frac{1}{2} \cdot 2} = {(- 6)}^{2}$

Step 1.2.4.2.1.3.2.2

Rewrite the expression.

$16 {(x + 5)}^{1} = {(- 6)}^{2}$

Step 1.2.4.2.1.4

Simplify.

$16 (x + 5) = {(- 6)}^{2}$

Step 1.2.4.2.1.5

Apply the distributive property.

$16 x + 16 \cdot 5 = {(- 6)}^{2}$

Step 1.2.4.2.1.6

Multiply $16$ by $5$ .

$16 x + 80 = {(- 6)}^{2}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Raise $- 6$ to the power of $2$ .

$16 x + 80 = 36$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

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

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Step 1.2.5.1.1

Subtract $80$ from both sides of the equation.

$16 x = 36 - 80$

Step 1.2.5.1.2

Subtract $80$ from $36$ .

$16 x = - 44$

Step 1.2.5.2

Divide each term in $16 x = - 44$ by $16$ and simplify.

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Step 1.2.5.2.1

Divide each term in $16 x = - 44$ by $16$ .

$\frac{16 x}{16} = \frac{- 44}{16}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor of $16$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{16 x}{16} = \frac{- 44}{16}$

Step 1.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 44}{16}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Cancel the common factor of $- 44$ and $16$ .

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Step 1.2.5.2.3.1.1

Factor $4$ out of $- 44$ .

$x = \frac{4 (- 11)}{16}$

Step 1.2.5.2.3.1.2

Cancel the common factors.

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Step 1.2.5.2.3.1.2.1

Factor $4$ out of $16$ .

$x = \frac{4 \cdot - 11}{4 \cdot 4}$

Step 1.2.5.2.3.1.2.2

Cancel the common factor.

$x = \frac{4 \cdot - 11}{4 \cdot 4}$

Step 1.2.5.2.3.1.2.3

Rewrite the expression.

$x = \frac{- 11}{4}$

Step 1.2.5.2.3.2

Move the negative in front of the fraction.

$x = - \frac{11}{4}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{11}{4}, 0)$

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 $y$ .

$y = - 4 \sqrt{(0) + 5} + 6$

Step 2.2

Add $0$ and $5$ .

$y = - 4 \sqrt{5} + 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 4 \sqrt{5} + 6)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{11}{4}, 0)$

y-intercept(s): $(0, - 4 \sqrt{5} + 6)$

Step 4