Finite Math Examples

$\sqrt{x + 150} - \sqrt{x + 55} = 5$

Step 1

Add $\sqrt{x + 55}$ to both sides of the equation.

$\sqrt{x + 150} = 5 + \sqrt{x + 55}$

Step 2

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

${\sqrt{x + 150}}^{2} = {(5 + \sqrt{x + 55})}^{2}$

Step 3

Simplify each side of the equation.

Tap for more steps...

Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 150}$ as ${(x + 150)}^{\frac{1}{2}}$ .

${({(x + 150)}^{\frac{1}{2}})}^{2} = {(5 + \sqrt{x + 55})}^{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${({(x + 150)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1

Multiply the exponents in ${({(x + 150)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1.1

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

${(x + 150)}^{\frac{1}{2} \cdot 2} = {(5 + \sqrt{x + 55})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

${(x + 150)}^{\frac{1}{2} \cdot 2} = {(5 + \sqrt{x + 55})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x + 150)}^{1} = {(5 + \sqrt{x + 55})}^{2}$

Step 3.2.1.2

Simplify.

$x + 150 = {(5 + \sqrt{x + 55})}^{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify ${(5 + \sqrt{x + 55})}^{2}$ .

Tap for more steps...

Step 3.3.1.1

Rewrite ${(5 + \sqrt{x + 55})}^{2}$ as $(5 + \sqrt{x + 55}) (5 + \sqrt{x + 55})$ .

$x + 150 = (5 + \sqrt{x + 55}) (5 + \sqrt{x + 55})$

Step 3.3.1.2

Expand $(5 + \sqrt{x + 55}) (5 + \sqrt{x + 55})$ using the FOIL Method.

Tap for more steps...

Step 3.3.1.2.1

Apply the distributive property.

$x + 150 = 5 (5 + \sqrt{x + 55}) + \sqrt{x + 55} (5 + \sqrt{x + 55})$

Step 3.3.1.2.2

Apply the distributive property.

$x + 150 = 5 \cdot 5 + 5 \sqrt{x + 55} + \sqrt{x + 55} (5 + \sqrt{x + 55})$

Step 3.3.1.2.3

Apply the distributive property.

$x + 150 = 5 \cdot 5 + 5 \sqrt{x + 55} + \sqrt{x + 55} \cdot 5 + \sqrt{x + 55} \sqrt{x + 55}$

Step 3.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.3.1.1

Multiply $5$ by $5$ .

$x + 150 = 25 + 5 \sqrt{x + 55} + \sqrt{x + 55} \cdot 5 + \sqrt{x + 55} \sqrt{x + 55}$

Step 3.3.1.3.1.2

Move $5$ to the left of $\sqrt{x + 55}$ .

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \cdot \sqrt{x + 55} + \sqrt{x + 55} \sqrt{x + 55}$

Step 3.3.1.3.1.3

Multiply $\sqrt{x + 55} \sqrt{x + 55}$ .

Tap for more steps...

Step 3.3.1.3.1.3.1

Raise $\sqrt{x + 55}$ to the power of $1$ .

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + {\sqrt{x + 55}}^{1} \sqrt{x + 55}$

Step 3.3.1.3.1.3.2

Raise $\sqrt{x + 55}$ to the power of $1$ .

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + {\sqrt{x + 55}}^{1} {\sqrt{x + 55}}^{1}$

Step 3.3.1.3.1.3.3

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

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + {\sqrt{x + 55}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + {\sqrt{x + 55}}^{2}$

Step 3.3.1.3.1.4

Rewrite ${\sqrt{x + 55}}^{2}$ as $x + 55$ .

Tap for more steps...

Step 3.3.1.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 55}$ as ${(x + 55)}^{\frac{1}{2}}$ .

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + {({(x + 55)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + {(x + 55)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

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

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + {(x + 55)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + {(x + 55)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + {(x + 55)}^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$x + 150 = 25 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + x + 55$

Step 3.3.1.3.2

Add $25$ and $55$ .

$x + 150 = 80 + 5 \sqrt{x + 55} + 5 \sqrt{x + 55} + x$

Step 3.3.1.3.3

Add $5 \sqrt{x + 55}$ and $5 \sqrt{x + 55}$ .

$x + 150 = 80 + 10 \sqrt{x + 55} + x$

Step 4

Solve for $10 \sqrt{x + 55}$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $80 + 10 \sqrt{x + 55} + x = x + 150$ .

$80 + 10 \sqrt{x + 55} + x = x + 150$

Step 4.2

Move all terms not containing $10 \sqrt{x + 55}$ to the right side of the equation.

Tap for more steps...

Step 4.2.1

Subtract $80$ from both sides of the equation.

$10 \sqrt{x + 55} + x = x + 150 - 80$

Step 4.2.2

Subtract $x$ from both sides of the equation.

$10 \sqrt{x + 55} = x + 150 - 80 - x$

Step 4.2.3

Combine the opposite terms in $x + 150 - 80 - x$ .

Tap for more steps...

Step 4.2.3.1

Subtract $x$ from $x$ .

$10 \sqrt{x + 55} = 0 + 150 - 80$

Step 4.2.3.2

Add $0$ and $150$ .

$10 \sqrt{x + 55} = 150 - 80$

Step 4.2.4

Subtract $80$ from $150$ .

$10 \sqrt{x + 55} = 70$

Step 5

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

${(10 \sqrt{x + 55})}^{2} = 70^{2}$

Step 6

Simplify each side of the equation.

Tap for more steps...

Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 55}$ as ${(x + 55)}^{\frac{1}{2}}$ .

${(10 {(x + 55)}^{\frac{1}{2}})}^{2} = 70^{2}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Simplify ${(10 {(x + 55)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 6.2.1.1

Apply the product rule to $10 {(x + 55)}^{\frac{1}{2}}$ .

$10^{2} {({(x + 55)}^{\frac{1}{2}})}^{2} = 70^{2}$

Step 6.2.1.2

Raise $10$ to the power of $2$ .

$100 {({(x + 55)}^{\frac{1}{2}})}^{2} = 70^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(x + 55)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 6.2.1.3.1

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

$100 {(x + 55)}^{\frac{1}{2} \cdot 2} = 70^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.1.3.2.1

Cancel the common factor.

$100 {(x + 55)}^{\frac{1}{2} \cdot 2} = 70^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$100 {(x + 55)}^{1} = 70^{2}$

Step 6.2.1.4

Simplify.

$100 (x + 55) = 70^{2}$

Step 6.2.1.5

Apply the distributive property.

$100 x + 100 \cdot 55 = 70^{2}$

Step 6.2.1.6

Multiply $100$ by $55$ .

$100 x + 5500 = 70^{2}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Raise $70$ to the power of $2$ .

$100 x + 5500 = 4900$

Step 7

Solve for $x$ .

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

Subtract $5500$ from both sides of the equation.

$100 x = 4900 - 5500$

Step 7.1.2

Subtract $5500$ from $4900$ .

$100 x = - 600$

Step 7.2

Divide each term in $100 x = - 600$ by $100$ and simplify.

Tap for more steps...

Step 7.2.1

Divide each term in $100 x = - 600$ by $100$ .

$\frac{100 x}{100} = \frac{- 600}{100}$

Step 7.2.2

Simplify the left side.

Tap for more steps...

Step 7.2.2.1

Cancel the common factor of $100$ .

Tap for more steps...

Step 7.2.2.1.1

Cancel the common factor.

$\frac{100 x}{100} = \frac{- 600}{100}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 600}{100}$

Step 7.2.3

Simplify the right side.

Tap for more steps...

Step 7.2.3.1

Divide $- 600$ by $100$ .

$x = - 6$