Algebra Examples

$f (x) = - \frac{1}{{(x + 3)}^{2}} + 2$

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 $x$ .

$0 = - \frac{1}{{(x + 3)}^{2}} + 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{1}{{(x + 3)}^{2}} + 2 = 0$ .

$- \frac{1}{{(x + 3)}^{2}} + 2 = 0$

Step 1.2.2

Subtract $2$ from both sides of the equation.

$- \frac{1}{{(x + 3)}^{2}} = - 2$

Step 1.2.3

Find the LCD of the terms in the equation.

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Step 1.2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

${(x + 3)}^{2}, 1$

Step 1.2.3.2

The LCM of one and any expression is the expression.

${(x + 3)}^{2}$

Step 1.2.4

Multiply each term in $- \frac{1}{{(x + 3)}^{2}} = - 2$ by ${(x + 3)}^{2}$ to eliminate the fractions.

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Step 1.2.4.1

Multiply each term in $- \frac{1}{{(x + 3)}^{2}} = - 2$ by ${(x + 3)}^{2}$ .

$- \frac{1}{{(x + 3)}^{2}} {(x + 3)}^{2} = - 2 {(x + 3)}^{2}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of ${(x + 3)}^{2}$ .

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Step 1.2.4.2.1.1

Move the leading negative in $- \frac{1}{{(x + 3)}^{2}}$ into the numerator.

$\frac{- 1}{{(x + 3)}^{2}} {(x + 3)}^{2} = - 2 {(x + 3)}^{2}$

Step 1.2.4.2.1.2

Cancel the common factor.

$\frac{- 1}{{(x + 3)}^{2}} {(x + 3)}^{2} = - 2 {(x + 3)}^{2}$

Step 1.2.4.2.1.3

Rewrite the expression.

$- 1 = - 2 {(x + 3)}^{2}$

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

Rewrite the equation as $- 2 {(x + 3)}^{2} = - 1$ .

$- 2 {(x + 3)}^{2} = - 1$

Step 1.2.5.2

Divide each term in $- 2 {(x + 3)}^{2} = - 1$ by $- 2$ and simplify.

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Step 1.2.5.2.1

Divide each term in $- 2 {(x + 3)}^{2} = - 1$ by $- 2$ .

$\frac{- 2 {(x + 3)}^{2}}{- 2} = \frac{- 1}{- 2}$

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 $- 2$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{- 2 {(x + 3)}^{2}}{- 2} = \frac{- 1}{- 2}$

Step 1.2.5.2.2.1.2

Divide ${(x + 3)}^{2}$ by $1$ .

${(x + 3)}^{2} = \frac{- 1}{- 2}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Dividing two negative values results in a positive value.

${(x + 3)}^{2} = \frac{1}{2}$

Step 1.2.5.3

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

$x + 3 = \pm \sqrt{\frac{1}{2}}$

Step 1.2.5.4

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

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Step 1.2.5.4.1

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

$x + 3 = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 1.2.5.4.2

Any root of $1$ is $1$ .

$x + 3 = \pm \frac{1}{\sqrt{2}}$

Step 1.2.5.4.3

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

$x + 3 = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.2.5.4.4

Combine and simplify the denominator.

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Step 1.2.5.4.4.1

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

$x + 3 = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.2.5.4.4.2

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

$x + 3 = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.2.5.4.4.3

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

$x + 3 = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.2.5.4.4.4

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

$x + 3 = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.5.4.4.5

Add $1$ and $1$ .

$x + 3 = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.2.5.4.4.6

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

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Step 1.2.5.4.4.6.1

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

$x + 3 = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.2.5.4.4.6.2

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

$x + 3 = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.4.4.6.3

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

$x + 3 = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.5.4.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.4.4.6.4.1

Cancel the common factor.

$x + 3 = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.5.4.4.6.4.2

Rewrite the expression.

$x + 3 = \pm \frac{\sqrt{2}}{2^{1}}$

Step 1.2.5.4.4.6.5

Evaluate the exponent.

$x + 3 = \pm \frac{\sqrt{2}}{2}$

Step 1.2.5.5

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

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Step 1.2.5.5.1

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

$x + 3 = \frac{\sqrt{2}}{2}$

Step 1.2.5.5.2

Subtract $3$ from both sides of the equation.

$x = \frac{\sqrt{2}}{2} - 3$

Step 1.2.5.5.3

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

$x + 3 = - \frac{\sqrt{2}}{2}$

Step 1.2.5.5.4

Subtract $3$ from both sides of the equation.

$x = - \frac{\sqrt{2}}{2} - 3$

Step 1.2.5.5.5

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

$x = \frac{\sqrt{2}}{2} - 3, - \frac{\sqrt{2}}{2} - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\sqrt{2}}{2} - 3, 0), (- \frac{\sqrt{2}}{2} - 3, 0)$

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 $y$ .

$y = - \frac{1}{{((0) + 3)}^{2}} + 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - \frac{1}{{(0 + 3)}^{2}} + 2$

Step 2.2.2

Remove parentheses.

$y = - \frac{1}{{((0) + 3)}^{2}} + 2$

Step 2.2.3

Simplify $- \frac{1}{{((0) + 3)}^{2}} + 2$ .

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Step 2.2.3.1

Simplify the denominator.

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Step 2.2.3.1.1

Add $0$ and $3$ .

$y = - \frac{1}{3^{2}} + 2$

Step 2.2.3.1.2

Raise $3$ to the power of $2$ .

$y = - \frac{1}{9} + 2$

Step 2.2.3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = - \frac{1}{9} + 2 \cdot \frac{9}{9}$

Step 2.2.3.3

Combine $2$ and $\frac{9}{9}$ .

$y = - \frac{1}{9} + \frac{2 \cdot 9}{9}$

Step 2.2.3.4

Combine the numerators over the common denominator.

$y = \frac{- 1 + 2 \cdot 9}{9}$

Step 2.2.3.5

Simplify the numerator.

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Step 2.2.3.5.1

Multiply $2$ by $9$ .

$y = \frac{- 1 + 18}{9}$

Step 2.2.3.5.2

Add $- 1$ and $18$ .

$y = \frac{17}{9}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{17}{9})$

Step 3

List the intersections.

x-intercept(s): $(\frac{\sqrt{2}}{2} - 3, 0), (- \frac{\sqrt{2}}{2} - 3, 0)$

y-intercept(s): $(0, \frac{17}{9})$

Step 4