Precalculus Examples

$9 {(x + 1)}^{\frac{1}{2}} - 10 {(x + 1)}^{\frac{3}{2}} + {(x + 1)}^{\frac{5}{2}} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor ${(x + 1)}^{\frac{1}{2}}$ out of $9 {(x + 1)}^{\frac{1}{2}} - 10 {(x + 1)}^{\frac{3}{2}} + {(x + 1)}^{\frac{5}{2}}$ .

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Step 1.1.1

Factor ${(x + 1)}^{\frac{1}{2}}$ out of $9 {(x + 1)}^{\frac{1}{2}}$ .

${(x + 1)}^{\frac{1}{2}} \cdot 9 - 10 {(x + 1)}^{\frac{3}{2}} + {(x + 1)}^{\frac{5}{2}} = 0$

Step 1.1.2

Factor ${(x + 1)}^{\frac{1}{2}}$ out of $- 10 {(x + 1)}^{\frac{3}{2}}$ .

${(x + 1)}^{\frac{1}{2}} \cdot 9 + {(x + 1)}^{\frac{1}{2}} (- 10 {(x + 1)}^{\frac{2}{2}}) + {(x + 1)}^{\frac{5}{2}} = 0$

Step 1.1.3

Factor ${(x + 1)}^{\frac{1}{2}}$ out of ${(x + 1)}^{\frac{5}{2}}$ .

${(x + 1)}^{\frac{1}{2}} \cdot 9 + {(x + 1)}^{\frac{1}{2}} (- 10 {(x + 1)}^{\frac{2}{2}}) + {(x + 1)}^{\frac{1}{2}} {(x + 1)}^{\frac{4}{2}} = 0$

Step 1.1.4

Factor ${(x + 1)}^{\frac{1}{2}}$ out of ${(x + 1)}^{\frac{1}{2}} \cdot 9 + {(x + 1)}^{\frac{1}{2}} (- 10 {(x + 1)}^{\frac{2}{2}})$ .

${(x + 1)}^{\frac{1}{2}} \cdot (9 - 10 {(x + 1)}^{\frac{2}{2}}) + {(x + 1)}^{\frac{1}{2}} {(x + 1)}^{\frac{4}{2}} = 0$

Step 1.1.5

Factor ${(x + 1)}^{\frac{1}{2}}$ out of ${(x + 1)}^{\frac{1}{2}} \cdot (9 - 10 {(x + 1)}^{\frac{2}{2}}) + {(x + 1)}^{\frac{1}{2}} {(x + 1)}^{\frac{4}{2}}$ .

${(x + 1)}^{\frac{1}{2}} (9 - 10 {(x + 1)}^{\frac{2}{2}} + {(x + 1)}^{\frac{4}{2}}) = 0$

Step 1.2

Factor $9 - 10 {(x + 1)}^{\frac{2}{2}} + {(x + 1)}^{\frac{4}{2}}$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $9$ and whose sum is $- 10$ .

$- 9, - 1$

Step 1.2.2

Write the factored form using these integers.

${(x + 1)}^{\frac{1}{2}} (({(x + 1)}^{\frac{2}{2}} - 9) ({(x + 1)}^{\frac{2}{2}} - 1)) = 0$

Step 1.3

Rewrite ${(x + 1)}^{\frac{2}{2}}$ as ${({(x + 1)}^{\frac{1}{2}})}^{2}$ .

${(x + 1)}^{\frac{1}{2}} (({({(x + 1)}^{\frac{1}{2}})}^{2} - 9) ({(x + 1)}^{\frac{2}{2}} - 1)) = 0$

Step 1.4

Rewrite $9$ as $3^{2}$ .

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

Step 1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x + 1)}^{\frac{1}{2}}$ and $b = 3$ .

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

Step 1.6

Rewrite ${(x + 1)}^{\frac{2}{2}}$ as ${({(x + 1)}^{\frac{1}{2}})}^{2}$ .

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

Step 1.7

Rewrite $1$ as $1^{2}$ .

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

Step 1.8

Factor.

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Step 1.8.1

Factor.

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Step 1.8.1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x + 1)}^{\frac{1}{2}}$ and $b = 1$ .

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

Step 1.8.1.2

Remove unnecessary parentheses.

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

Step 1.8.2

Remove unnecessary parentheses.

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

Step 2

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

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

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

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

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

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

Step 3

Set ${(x + 1)}^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set ${(x + 1)}^{\frac{1}{2}}$ equal to $0$ .

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

Step 3.2

Solve ${(x + 1)}^{\frac{1}{2}} = 0$ for $x$ .

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Step 3.2.1

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

Set ${(x + 1)}^{\frac{1}{2}} + 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set ${(x + 1)}^{\frac{1}{2}} + 3$ equal to $0$ .

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

Step 4.2

Solve ${(x + 1)}^{\frac{1}{2}} + 3 = 0$ for $x$ .

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Step 4.2.1

Subtract $3$ from both sides of the equation.

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

Step 4.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 4.2.3

Simplify the exponent.

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Step 4.2.3.1

Simplify the left side.

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Step 4.2.3.1.1

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

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Step 4.2.3.1.1.1

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

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Step 4.2.3.1.1.1.1

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

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

Step 4.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 4.2.3.1.1.2

Simplify.

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

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

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

$x + 1 = 9$

Step 4.2.4

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

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Step 4.2.4.1

Subtract $1$ from both sides of the equation.

$x = 9 - 1$

Step 4.2.4.2

Subtract $1$ from $9$ .

$x = 8$

Step 5

Set ${(x + 1)}^{\frac{1}{2}} + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set ${(x + 1)}^{\frac{1}{2}} + 1$ equal to $0$ .

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

Step 5.2

Solve ${(x + 1)}^{\frac{1}{2}} + 1 = 0$ for $x$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

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

Step 5.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 5.2.3

Simplify the exponent.

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Step 5.2.3.1

Simplify the left side.

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Step 5.2.3.1.1

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

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Step 5.2.3.1.1.1

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

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Step 5.2.3.1.1.1.1

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

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

Step 5.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 5.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 5.2.3.1.1.2

Simplify.

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

Step 5.2.3.2

Simplify the right side.

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Step 5.2.3.2.1

Raise $- 1$ to the power of $2$ .

$x + 1 = 1$

Step 5.2.4

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

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Step 5.2.4.1

Subtract $1$ from both sides of the equation.

$x = 1 - 1$

Step 5.2.4.2

Subtract $1$ from $1$ .

$x = 0$

Step 6

The final solution is all the values that make ${(x + 1)}^{\frac{1}{2}} ({(x + 1)}^{\frac{1}{2}} + 3) ({(x + 1)}^{\frac{1}{2}} - 3) ({(x + 1)}^{\frac{1}{2}} + 1) ({(x + 1)}^{\frac{1}{2}} - 1) = 0$ true.

$x = - 1, 8, 0$