Finite Math Examples

$\sqrt{2 x + 169} - \sqrt{x + 1} = 12$

Step 1

Add $\sqrt{x + 1}$ to both sides of the equation.

$\sqrt{2 x + 169} = 12 + \sqrt{x + 1}$

Step 2

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

${\sqrt{2 x + 169}}^{2} = {(12 + \sqrt{x + 1})}^{2}$

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 $\sqrt{2 x + 169}$ as ${(2 x + 169)}^{\frac{1}{2}}$ .

${({(2 x + 169)}^{\frac{1}{2}})}^{2} = {(12 + \sqrt{x + 1})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(2 x + 169)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(2 x + 169)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(2 x + 169)}^{\frac{1}{2} \cdot 2} = {(12 + \sqrt{x + 1})}^{2}$

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.

${(2 x + 169)}^{\frac{1}{2} \cdot 2} = {(12 + \sqrt{x + 1})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x + 169)}^{1} = {(12 + \sqrt{x + 1})}^{2}$

Step 3.2.1.2

Simplify.

$2 x + 169 = {(12 + \sqrt{x + 1})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(12 + \sqrt{x + 1})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(12 + \sqrt{x + 1})}^{2}$ as $(12 + \sqrt{x + 1}) (12 + \sqrt{x + 1})$ .

$2 x + 169 = (12 + \sqrt{x + 1}) (12 + \sqrt{x + 1})$

Step 3.3.1.2

Expand $(12 + \sqrt{x + 1}) (12 + \sqrt{x + 1})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$2 x + 169 = 12 (12 + \sqrt{x + 1}) + \sqrt{x + 1} (12 + \sqrt{x + 1})$

Step 3.3.1.2.2

Apply the distributive property.

$2 x + 169 = 12 \cdot 12 + 12 \sqrt{x + 1} + \sqrt{x + 1} (12 + \sqrt{x + 1})$

Step 3.3.1.2.3

Apply the distributive property.

$2 x + 169 = 12 \cdot 12 + 12 \sqrt{x + 1} + \sqrt{x + 1} \cdot 12 + \sqrt{x + 1} \sqrt{x + 1}$

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 $12$ by $12$ .

$2 x + 169 = 144 + 12 \sqrt{x + 1} + \sqrt{x + 1} \cdot 12 + \sqrt{x + 1} \sqrt{x + 1}$

Step 3.3.1.3.1.2

Move $12$ to the left of $\sqrt{x + 1}$ .

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \cdot \sqrt{x + 1} + \sqrt{x + 1} \sqrt{x + 1}$

Step 3.3.1.3.1.3

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

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Step 3.3.1.3.1.3.1

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

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + {\sqrt{x + 1}}^{1} \sqrt{x + 1}$

Step 3.3.1.3.1.3.2

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

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + {\sqrt{x + 1}}^{1} {\sqrt{x + 1}}^{1}$

Step 3.3.1.3.1.3.3

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

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + {\sqrt{x + 1}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + {\sqrt{x + 1}}^{2}$

Step 3.3.1.3.1.4

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

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Step 3.3.1.3.1.4.1

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

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + {({(x + 1)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + {(x + 1)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

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

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + {(x + 1)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + {(x + 1)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + {(x + 1)}^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$2 x + 169 = 144 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + x + 1$

Step 3.3.1.3.2

Add $144$ and $1$ .

$2 x + 169 = 145 + 12 \sqrt{x + 1} + 12 \sqrt{x + 1} + x$

Step 3.3.1.3.3

Add $12 \sqrt{x + 1}$ and $12 \sqrt{x + 1}$ .

$2 x + 169 = 145 + 24 \sqrt{x + 1} + x$

Step 4

Solve for $24 \sqrt{x + 1}$ .

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Step 4.1

Rewrite the equation as $145 + 24 \sqrt{x + 1} + x = 2 x + 169$ .

$145 + 24 \sqrt{x + 1} + x = 2 x + 169$

Step 4.2

Move all terms not containing $24 \sqrt{x + 1}$ to the right side of the equation.

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Step 4.2.1

Subtract $145$ from both sides of the equation.

$24 \sqrt{x + 1} + x = 2 x + 169 - 145$

Step 4.2.2

Subtract $x$ from both sides of the equation.

$24 \sqrt{x + 1} = 2 x + 169 - 145 - x$

Step 4.2.3

Subtract $x$ from $2 x$ .

$24 \sqrt{x + 1} = x + 169 - 145$

Step 4.2.4

Subtract $145$ from $169$ .

$24 \sqrt{x + 1} = x + 24$

Step 5

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

${(24 \sqrt{x + 1})}^{2} = {(x + 24)}^{2}$

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

${(24 {(x + 1)}^{\frac{1}{2}})}^{2} = {(x + 24)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(24 {(x + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $24 {(x + 1)}^{\frac{1}{2}}$ .

$24^{2} {({(x + 1)}^{\frac{1}{2}})}^{2} = {(x + 24)}^{2}$

Step 6.2.1.2

Raise $24$ to the power of $2$ .

$576 {({(x + 1)}^{\frac{1}{2}})}^{2} = {(x + 24)}^{2}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$576 {(x + 1)}^{\frac{1}{2} \cdot 2} = {(x + 24)}^{2}$

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.

$576 {(x + 1)}^{\frac{1}{2} \cdot 2} = {(x + 24)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$576 {(x + 1)}^{1} = {(x + 24)}^{2}$

Step 6.2.1.4

Simplify.

$576 (x + 1) = {(x + 24)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$576 x + 576 \cdot 1 = {(x + 24)}^{2}$

Step 6.2.1.6

Multiply $576$ by $1$ .

$576 x + 576 = {(x + 24)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(x + 24)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(x + 24)}^{2}$ as $(x + 24) (x + 24)$ .

$576 x + 576 = (x + 24) (x + 24)$

Step 6.3.1.2

Expand $(x + 24) (x + 24)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$576 x + 576 = x (x + 24) + 24 (x + 24)$

Step 6.3.1.2.2

Apply the distributive property.

$576 x + 576 = x \cdot x + x \cdot 24 + 24 (x + 24)$

Step 6.3.1.2.3

Apply the distributive property.

$576 x + 576 = x \cdot x + x \cdot 24 + 24 x + 24 \cdot 24$

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 $x$ .

$576 x + 576 = x^{2} + x \cdot 24 + 24 x + 24 \cdot 24$

Step 6.3.1.3.1.2

Move $24$ to the left of $x$ .

$576 x + 576 = x^{2} + 24 \cdot x + 24 x + 24 \cdot 24$

Step 6.3.1.3.1.3

Multiply $24$ by $24$ .

$576 x + 576 = x^{2} + 24 x + 24 x + 576$

Step 6.3.1.3.2

Add $24 x$ and $24 x$ .

$576 x + 576 = x^{2} + 48 x + 576$

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.

$x^{2} + 48 x + 576 = 576 x + 576$

Step 7.2

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

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Step 7.2.1

Subtract $576 x$ from both sides of the equation.

$x^{2} + 48 x + 576 - 576 x = 576$

Step 7.2.2

Subtract $576 x$ from $48 x$ .

$x^{2} - 528 x + 576 = 576$

Step 7.3

Subtract $576$ from both sides of the equation.

$x^{2} - 528 x + 576 - 576 = 0$

Step 7.4

Combine the opposite terms in $x^{2} - 528 x + 576 - 576$ .

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Step 7.4.1

Subtract $576$ from $576$ .

$x^{2} - 528 x + 0 = 0$

Step 7.4.2

Add $x^{2} - 528 x$ and $0$ .

$x^{2} - 528 x = 0$

Step 7.5

Factor $x$ out of $x^{2} - 528 x$ .

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Step 7.5.1

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

$x \cdot x - 528 x = 0$

Step 7.5.2

Factor $x$ out of $- 528 x$ .

$x \cdot x + x \cdot - 528 = 0$

Step 7.5.3

Factor $x$ out of $x \cdot x + x \cdot - 528$ .

$x (x - 528) = 0$

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 $0$ .

$x = 0$

$x - 528 = 0$

Step 7.7

Set $x$ equal to $0$ .

$x = 0$

Step 7.8

Set $x - 528$ equal to $0$ and solve for $x$ .

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Step 7.8.1

Set $x - 528$ equal to $0$ .

$x - 528 = 0$

Step 7.8.2

Add $528$ to both sides of the equation.

$x = 528$

Step 7.9

The final solution is all the values that make $x (x - 528) = 0$ true.

$x = 0, 528$

Step 8