Pre-Algebra Examples

$\sqrt{2 x + 5} - \sqrt{x + 6} = 1$

Step 1

Add $\sqrt{x + 6}$ to both sides of the equation.

$\sqrt{2 x + 5} = 1 + \sqrt{x + 6}$

Step 2

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

${\sqrt{2 x + 5}}^{2} = {(1 + \sqrt{x + 6})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x + 5)}^{1} = {(1 + \sqrt{x + 6})}^{2}$

Step 3.2.1.2

Simplify.

$2 x + 5 = {(1 + \sqrt{x + 6})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(1 + \sqrt{x + 6})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(1 + \sqrt{x + 6})}^{2}$ as $(1 + \sqrt{x + 6}) (1 + \sqrt{x + 6})$ .

$2 x + 5 = (1 + \sqrt{x + 6}) (1 + \sqrt{x + 6})$

Step 3.3.1.2

Expand $(1 + \sqrt{x + 6}) (1 + \sqrt{x + 6})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$2 x + 5 = 1 (1 + \sqrt{x + 6}) + \sqrt{x + 6} (1 + \sqrt{x + 6})$

Step 3.3.1.2.2

Apply the distributive property.

$2 x + 5 = 1 \cdot 1 + 1 \sqrt{x + 6} + \sqrt{x + 6} (1 + \sqrt{x + 6})$

Step 3.3.1.2.3

Apply the distributive property.

$2 x + 5 = 1 \cdot 1 + 1 \sqrt{x + 6} + \sqrt{x + 6} \cdot 1 + \sqrt{x + 6} \sqrt{x + 6}$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $1$ by $1$ .

$2 x + 5 = 1 + 1 \sqrt{x + 6} + \sqrt{x + 6} \cdot 1 + \sqrt{x + 6} \sqrt{x + 6}$

Step 3.3.1.3.1.2

Multiply $\sqrt{x + 6}$ by $1$ .

$2 x + 5 = 1 + \sqrt{x + 6} + \sqrt{x + 6} \cdot 1 + \sqrt{x + 6} \sqrt{x + 6}$

Step 3.3.1.3.1.3

Multiply $\sqrt{x + 6}$ by $1$ .

$2 x + 5 = 1 + \sqrt{x + 6} + \sqrt{x + 6} + \sqrt{x + 6} \sqrt{x + 6}$

Step 3.3.1.3.1.4

Multiply $\sqrt{x + 6} \sqrt{x + 6}$ .

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Step 3.3.1.3.1.4.1

Raise $\sqrt{x + 6}$ to the power of $1$ .

$2 x + 5 = 1 + \sqrt{x + 6} + \sqrt{x + 6} + {\sqrt{x + 6}}^{1} \sqrt{x + 6}$

Step 3.3.1.3.1.4.2

Raise $\sqrt{x + 6}$ to the power of $1$ .

$2 x + 5 = 1 + \sqrt{x + 6} + \sqrt{x + 6} + {\sqrt{x + 6}}^{1} {\sqrt{x + 6}}^{1}$

Step 3.3.1.3.1.4.3

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

$2 x + 5 = 1 + \sqrt{x + 6} + \sqrt{x + 6} + {\sqrt{x + 6}}^{1 + 1}$

Step 3.3.1.3.1.4.4

Add $1$ and $1$ .

$2 x + 5 = 1 + \sqrt{x + 6} + \sqrt{x + 6} + {\sqrt{x + 6}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{x + 6}}^{2}$ as $x + 6$ .

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Step 3.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 6}$ as ${(x + 6)}^{\frac{1}{2}}$ .

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

Step 3.3.1.3.1.5.2

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

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

Step 3.3.1.3.1.5.3

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

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

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

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

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$2 x + 5 = 1 + \sqrt{x + 6} + \sqrt{x + 6} + {(x + 6)}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$2 x + 5 = 1 + \sqrt{x + 6} + \sqrt{x + 6} + x + 6$

Step 3.3.1.3.2

Add $1$ and $6$ .

$2 x + 5 = 7 + \sqrt{x + 6} + \sqrt{x + 6} + x$

Step 3.3.1.3.3

Add $\sqrt{x + 6}$ and $\sqrt{x + 6}$ .

$2 x + 5 = 7 + 2 \sqrt{x + 6} + x$

Step 4

Solve for $2 \sqrt{x + 6}$ .

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Step 4.1

Rewrite the equation as $7 + 2 \sqrt{x + 6} + x = 2 x + 5$ .

$7 + 2 \sqrt{x + 6} + x = 2 x + 5$

Step 4.2

Move all terms not containing $2 \sqrt{x + 6}$ to the right side of the equation.

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Step 4.2.1

Subtract $7$ from both sides of the equation.

$2 \sqrt{x + 6} + x = 2 x + 5 - 7$

Step 4.2.2

Subtract $x$ from both sides of the equation.

$2 \sqrt{x + 6} = 2 x + 5 - 7 - x$

Step 4.2.3

Subtract $x$ from $2 x$ .

$2 \sqrt{x + 6} = x + 5 - 7$

Step 4.2.4

Subtract $7$ from $5$ .

$2 \sqrt{x + 6} = x - 2$

Step 5

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

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

Step 6

Simplify each side of the equation.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 6}$ as ${(x + 6)}^{\frac{1}{2}}$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(2 {(x + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $2 {(x + 6)}^{\frac{1}{2}}$ .

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

Step 6.2.1.2

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

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

Step 6.2.1.3

Multiply the exponents in ${({(x + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

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

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

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

Step 6.2.1.3.2.2

Rewrite the expression.

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

Step 6.2.1.4

Simplify.

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

Step 6.2.1.5

Apply the distributive property.

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

Step 6.2.1.6

Multiply $4$ by $6$ .

$4 x + 24 = {(x - 2)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(x - 2)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$4 x + 24 = (x - 2) (x - 2)$

Step 6.3.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$4 x + 24 = x (x - 2) - 2 (x - 2)$

Step 6.3.1.2.2

Apply the distributive property.

$4 x + 24 = x \cdot x + x \cdot - 2 - 2 (x - 2)$

Step 6.3.1.2.3

Apply the distributive property.

$4 x + 24 = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Multiply $x$ by $x$ .

$4 x + 24 = x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 6.3.1.3.1.2

Move $- 2$ to the left of $x$ .

$4 x + 24 = x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2$

Step 6.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

$4 x + 24 = x^{2} - 2 x - 2 x + 4$

Step 6.3.1.3.2

Subtract $2 x$ from $- 2 x$ .

$4 x + 24 = x^{2} - 4 x + 4$

Step 7

Solve for $x$ .

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Step 7.1

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

$x^{2} - 4 x + 4 = 4 x + 24$

Step 7.2

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

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Step 7.2.1

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

$x^{2} - 4 x + 4 - 4 x = 24$

Step 7.2.2

Subtract $4 x$ from $- 4 x$ .

$x^{2} - 8 x + 4 = 24$

Step 7.3

Subtract $24$ from both sides of the equation.

$x^{2} - 8 x + 4 - 24 = 0$

Step 7.4

Subtract $24$ from $4$ .

$x^{2} - 8 x - 20 = 0$

Step 7.5

Factor $x^{2} - 8 x - 20$ using the AC method.

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Step 7.5.1

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 $- 20$ and whose sum is $- 8$ .

$- 10, 2$

Step 7.5.2

Write the factored form using these integers.

$(x - 10) (x + 2) = 0$

Step 7.6

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

$x - 10 = 0$

$x + 2 = 0$

Step 7.7

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 7.7.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 7.7.2

Add $10$ to both sides of the equation.

$x = 10$

Step 7.8

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 7.8.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 7.8.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 7.9

The final solution is all the values that make $(x - 10) (x + 2) = 0$ true.

$x = 10, - 2$

Step 8

Exclude the solutions that do not make $\sqrt{2 x + 5} - \sqrt{x + 6} = 1$ true.

$x = 10$