Algebra Examples

$\sqrt{9 x + 81} = x + 5$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $x$ from both sides of the equation.

$\sqrt{9 x + 81} - x = 5$

Step 1.2

Subtract $5$ from both sides of the equation.

$\sqrt{9 x + 81} - x - 5 = 0$

Step 2

Simplify each term.

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Step 2.1

Factor $9$ out of $9 x + 81$ .

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Step 2.1.1

Factor $9$ out of $9 x$ .

$\sqrt{9 (x) + 81} - x - 5 = 0$

Step 2.1.2

Factor $9$ out of $81$ .

$\sqrt{9 x + 9 \cdot 9} - x - 5 = 0$

Step 2.1.3

Factor $9$ out of $9 x + 9 \cdot 9$ .

$\sqrt{9 (x + 9)} - x - 5 = 0$

Step 2.2

Rewrite $9 (x + 9)$ as $3^{2} (x + 3^{2})$ .

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Step 2.2.1

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

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

Step 2.2.2

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

$\sqrt{3^{2} (x + 3^{2})} - x - 5 = 0$

Step 2.3

Pull terms out from under the radical.

$3 \sqrt{x + 3^{2}} - x - 5 = 0$

Step 2.4

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

$3 \sqrt{x + 9} - x - 5 = 0$

Step 3

Move all terms not containing $3 \sqrt{x + 9}$ to the right side of the equation.

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Step 3.1

Add $x$ to both sides of the equation.

$3 \sqrt{x + 9} - 5 = x$

Step 3.2

Add $5$ to both sides of the equation.

$3 \sqrt{x + 9} = x + 5$

Step 4

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

${(3 \sqrt{x + 9})}^{2} = {(x + 5)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 9}$ as ${(x + 9)}^{\frac{1}{2}}$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(3 {(x + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Apply the product rule to $3 {(x + 9)}^{\frac{1}{2}}$ .

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

Step 5.2.1.2

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

$9 {({(x + 9)}^{\frac{1}{2}})}^{2} = {(x + 5)}^{2}$

Step 5.2.1.3

Multiply the exponents in ${({(x + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.3.1

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

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

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

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

Step 5.2.1.3.2.2

Rewrite the expression.

$9 {(x + 9)}^{1} = {(x + 5)}^{2}$

Step 5.2.1.4

Simplify.

$9 (x + 9) = {(x + 5)}^{2}$

Step 5.2.1.5

Apply the distributive property.

$9 x + 9 \cdot 9 = {(x + 5)}^{2}$

Step 5.2.1.6

Multiply $9$ by $9$ .

$9 x + 81 = {(x + 5)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(x + 5)}^{2}$ .

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Step 5.3.1.1

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$9 x + 81 = (x + 5) (x + 5)$

Step 5.3.1.2

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 5.3.1.2.1

Apply the distributive property.

$9 x + 81 = x (x + 5) + 5 (x + 5)$

Step 5.3.1.2.2

Apply the distributive property.

$9 x + 81 = x \cdot x + x \cdot 5 + 5 (x + 5)$

Step 5.3.1.2.3

Apply the distributive property.

$9 x + 81 = x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5$

Step 5.3.1.3

Simplify and combine like terms.

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Step 5.3.1.3.1

Simplify each term.

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Step 5.3.1.3.1.1

Multiply $x$ by $x$ .

$9 x + 81 = x^{2} + x \cdot 5 + 5 x + 5 \cdot 5$

Step 5.3.1.3.1.2

Move $5$ to the left of $x$ .

$9 x + 81 = x^{2} + 5 \cdot x + 5 x + 5 \cdot 5$

Step 5.3.1.3.1.3

Multiply $5$ by $5$ .

$9 x + 81 = x^{2} + 5 x + 5 x + 25$

Step 5.3.1.3.2

Add $5 x$ and $5 x$ .

$9 x + 81 = x^{2} + 10 x + 25$

Step 6

Solve for $x$ .

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Step 6.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 10 x + 25 = 9 x + 81$

Step 6.2

Move all terms containing $x$ to the left side of the equation.

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Step 6.2.1

Subtract $9 x$ from both sides of the equation.

$x^{2} + 10 x + 25 - 9 x = 81$

Step 6.2.2

Subtract $9 x$ from $10 x$ .

$x^{2} + x + 25 = 81$

Step 6.3

Subtract $81$ from both sides of the equation.

$x^{2} + x + 25 - 81 = 0$

Step 6.4

Subtract $81$ from $25$ .

$x^{2} + x - 56 = 0$

Step 6.5

Factor $x^{2} + x - 56$ using the AC method.

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Step 6.5.1

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 $- 56$ and whose sum is $1$ .

$- 7, 8$

Step 6.5.2

Write the factored form using these integers.

$(x - 7) (x + 8) = 0$

Step 6.6

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

$x - 7 = 0$

$x + 8 = 0$

Step 6.7

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 6.7.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 6.7.2

Add $7$ to both sides of the equation.

$x = 7$

Step 6.8

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 6.8.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 6.8.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 6.9

The final solution is all the values that make $(x - 7) (x + 8) = 0$ true.

$x = 7, - 8$

Step 7

Exclude the solutions that do not make $3 \sqrt{x + 9} - x - 5 = 0$ true.

$x = 7$