Pre-Algebra Examples

$\sqrt{3 x + 28} = 1 + \sqrt{x + 9}$

Step 1

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

${\sqrt{3 x + 28}}^{2} = {(1 + \sqrt{x + 9})}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 28}$ as ${(3 x + 28)}^{\frac{1}{2}}$ .

${({(3 x + 28)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{x + 9})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(3 x + 28)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(3 x + 28)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(3 x + 28)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{x + 9})}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(3 x + 28)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{x + 9})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(3 x + 28)}^{1} = {(1 + \sqrt{x + 9})}^{2}$

Step 2.2.1.2

Simplify.

$3 x + 28 = {(1 + \sqrt{x + 9})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite ${(1 + \sqrt{x + 9})}^{2}$ as $(1 + \sqrt{x + 9}) (1 + \sqrt{x + 9})$ .

$3 x + 28 = (1 + \sqrt{x + 9}) (1 + \sqrt{x + 9})$

Step 2.3.1.2

Expand $(1 + \sqrt{x + 9}) (1 + \sqrt{x + 9})$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$3 x + 28 = 1 (1 + \sqrt{x + 9}) + \sqrt{x + 9} (1 + \sqrt{x + 9})$

Step 2.3.1.2.2

Apply the distributive property.

$3 x + 28 = 1 \cdot 1 + 1 \sqrt{x + 9} + \sqrt{x + 9} (1 + \sqrt{x + 9})$

Step 2.3.1.2.3

Apply the distributive property.

$3 x + 28 = 1 \cdot 1 + 1 \sqrt{x + 9} + \sqrt{x + 9} \cdot 1 + \sqrt{x + 9} \sqrt{x + 9}$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $1$ by $1$ .

$3 x + 28 = 1 + 1 \sqrt{x + 9} + \sqrt{x + 9} \cdot 1 + \sqrt{x + 9} \sqrt{x + 9}$

Step 2.3.1.3.1.2

Multiply $\sqrt{x + 9}$ by $1$ .

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} \cdot 1 + \sqrt{x + 9} \sqrt{x + 9}$

Step 2.3.1.3.1.3

Multiply $\sqrt{x + 9}$ by $1$ .

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + \sqrt{x + 9} \sqrt{x + 9}$

Step 2.3.1.3.1.4

Multiply $\sqrt{x + 9} \sqrt{x + 9}$ .

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Step 2.3.1.3.1.4.1

Raise $\sqrt{x + 9}$ to the power of $1$ .

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + {\sqrt{x + 9}}^{1} \sqrt{x + 9}$

Step 2.3.1.3.1.4.2

Raise $\sqrt{x + 9}$ to the power of $1$ .

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + {\sqrt{x + 9}}^{1} {\sqrt{x + 9}}^{1}$

Step 2.3.1.3.1.4.3

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

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + {\sqrt{x + 9}}^{1 + 1}$

Step 2.3.1.3.1.4.4

Add $1$ and $1$ .

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + {\sqrt{x + 9}}^{2}$

Step 2.3.1.3.1.5

Rewrite ${\sqrt{x + 9}}^{2}$ as $x + 9$ .

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Step 2.3.1.3.1.5.1

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

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + {({(x + 9)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.3.1.5.2

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

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + {(x + 9)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3.1.5.3

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

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + {(x + 9)}^{\frac{2}{2}}$

Step 2.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.5.4.1

Cancel the common factor.

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + {(x + 9)}^{\frac{2}{2}}$

Step 2.3.1.3.1.5.4.2

Rewrite the expression.

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + {(x + 9)}^{1}$

Step 2.3.1.3.1.5.5

Simplify.

$3 x + 28 = 1 + \sqrt{x + 9} + \sqrt{x + 9} + x + 9$

Step 2.3.1.3.2

Add $1$ and $9$ .

$3 x + 28 = 10 + \sqrt{x + 9} + \sqrt{x + 9} + x$

Step 2.3.1.3.3

Add $\sqrt{x + 9}$ and $\sqrt{x + 9}$ .

$3 x + 28 = 10 + 2 \sqrt{x + 9} + x$

Step 3

Solve for $2 \sqrt{x + 9}$ .

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Step 3.1

Rewrite the equation as $10 + 2 \sqrt{x + 9} + x = 3 x + 28$ .

$10 + 2 \sqrt{x + 9} + x = 3 x + 28$

Step 3.2

Move all terms not containing $2 \sqrt{x + 9}$ to the right side of the equation.

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Step 3.2.1

Subtract $10$ from both sides of the equation.

$2 \sqrt{x + 9} + x = 3 x + 28 - 10$

Step 3.2.2

Subtract $x$ from both sides of the equation.

$2 \sqrt{x + 9} = 3 x + 28 - 10 - x$

Step 3.2.3

Subtract $x$ from $3 x$ .

$2 \sqrt{x + 9} = 2 x + 28 - 10$

Step 3.2.4

Subtract $10$ from $28$ .

$2 \sqrt{x + 9} = 2 x + 18$

Step 4

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

${(2 \sqrt{x + 9})}^{2} = {(2 x + 18)}^{2}$

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

${(2 {(x + 9)}^{\frac{1}{2}})}^{2} = {(2 x + 18)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

$2^{2} {({(x + 9)}^{\frac{1}{2}})}^{2} = {(2 x + 18)}^{2}$

Step 5.2.1.2

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

$4 {({(x + 9)}^{\frac{1}{2}})}^{2} = {(2 x + 18)}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$4 {(x + 9)}^{\frac{1}{2} \cdot 2} = {(2 x + 18)}^{2}$

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.

$4 {(x + 9)}^{\frac{1}{2} \cdot 2} = {(2 x + 18)}^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$4 {(x + 9)}^{1} = {(2 x + 18)}^{2}$

Step 5.2.1.4

Simplify.

$4 (x + 9) = {(2 x + 18)}^{2}$

Step 5.2.1.5

Apply the distributive property.

$4 x + 4 \cdot 9 = {(2 x + 18)}^{2}$

Step 5.2.1.6

Multiply $4$ by $9$ .

$4 x + 36 = {(2 x + 18)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(2 x + 18)}^{2}$ .

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Step 5.3.1.1

Rewrite ${(2 x + 18)}^{2}$ as $(2 x + 18) (2 x + 18)$ .

$4 x + 36 = (2 x + 18) (2 x + 18)$

Step 5.3.1.2

Expand $(2 x + 18) (2 x + 18)$ using the FOIL Method.

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Step 5.3.1.2.1

Apply the distributive property.

$4 x + 36 = 2 x (2 x + 18) + 18 (2 x + 18)$

Step 5.3.1.2.2

Apply the distributive property.

$4 x + 36 = 2 x (2 x) + 2 x \cdot 18 + 18 (2 x + 18)$

Step 5.3.1.2.3

Apply the distributive property.

$4 x + 36 = 2 x (2 x) + 2 x \cdot 18 + 18 (2 x) + 18 \cdot 18$

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

Rewrite using the commutative property of multiplication.

$4 x + 36 = 2 \cdot 2 x \cdot x + 2 x \cdot 18 + 18 (2 x) + 18 \cdot 18$

Step 5.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.1.3.1.2.1

Move $x$ .

$4 x + 36 = 2 \cdot 2 (x \cdot x) + 2 x \cdot 18 + 18 (2 x) + 18 \cdot 18$

Step 5.3.1.3.1.2.2

Multiply $x$ by $x$ .

$4 x + 36 = 2 \cdot 2 x^{2} + 2 x \cdot 18 + 18 (2 x) + 18 \cdot 18$

Step 5.3.1.3.1.3

Multiply $2$ by $2$ .

$4 x + 36 = 4 x^{2} + 2 x \cdot 18 + 18 (2 x) + 18 \cdot 18$

Step 5.3.1.3.1.4

Multiply $18$ by $2$ .

$4 x + 36 = 4 x^{2} + 36 x + 18 (2 x) + 18 \cdot 18$

Step 5.3.1.3.1.5

Multiply $2$ by $18$ .

$4 x + 36 = 4 x^{2} + 36 x + 36 x + 18 \cdot 18$

Step 5.3.1.3.1.6

Multiply $18$ by $18$ .

$4 x + 36 = 4 x^{2} + 36 x + 36 x + 324$

Step 5.3.1.3.2

Add $36 x$ and $36 x$ .

$4 x + 36 = 4 x^{2} + 72 x + 324$

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.

$4 x^{2} + 72 x + 324 = 4 x + 36$

Step 6.2

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

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Step 6.2.1

Subtract $4 x$ from both sides of the equation.

$4 x^{2} + 72 x + 324 - 4 x = 36$

Step 6.2.2

Subtract $4 x$ from $72 x$ .

$4 x^{2} + 68 x + 324 = 36$

Step 6.3

Subtract $36$ from both sides of the equation.

$4 x^{2} + 68 x + 324 - 36 = 0$

Step 6.4

Subtract $36$ from $324$ .

$4 x^{2} + 68 x + 288 = 0$

Step 6.5

Factor the left side of the equation.

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Step 6.5.1

Factor $4$ out of $4 x^{2} + 68 x + 288$ .

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Step 6.5.1.1

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) + 68 x + 288 = 0$

Step 6.5.1.2

Factor $4$ out of $68 x$ .

$4 (x^{2}) + 4 (17 x) + 288 = 0$

Step 6.5.1.3

Factor $4$ out of $288$ .

$4 x^{2} + 4 (17 x) + 4 \cdot 72 = 0$

Step 6.5.1.4

Factor $4$ out of $4 x^{2} + 4 (17 x)$ .

$4 (x^{2} + 17 x) + 4 \cdot 72 = 0$

Step 6.5.1.5

Factor $4$ out of $4 (x^{2} + 17 x) + 4 \cdot 72$ .

$4 (x^{2} + 17 x + 72) = 0$

Step 6.5.2

Factor.

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Step 6.5.2.1

Factor $x^{2} + 17 x + 72$ using the AC method.

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Step 6.5.2.1.1

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 $72$ and whose sum is $17$ .

$8, 9$

Step 6.5.2.1.2

Write the factored form using these integers.

$4 ((x + 8) (x + 9)) = 0$

Step 6.5.2.2

Remove unnecessary parentheses.

$4 (x + 8) (x + 9) = 0$

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 $0$ .

$x + 8 = 0$

$x + 9 = 0$

Step 6.7

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

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Step 6.7.1

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

$x + 8 = 0$

Step 6.7.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 6.8

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 6.8.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 6.8.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 6.9

The final solution is all the values that make $4 (x + 8) (x + 9) = 0$ true.

$x = - 8, - 9$