Finite Math Examples

$x - 4 = \sqrt{9 x + 16}$

Step 1

Subtract $\sqrt{9 x + 16}$ from both sides of the equation.

$x - 4 - \sqrt{9 x + 16} = 0$

Step 2

Move all terms not containing $- \sqrt{9 x + 16}$ to the right side of the equation.

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Step 2.1

Subtract $x$ from both sides of the equation.

$- 4 - \sqrt{9 x + 16} = - x$

Step 2.2

Add $4$ to both sides of the equation.

$- \sqrt{9 x + 16} = - x + 4$

Step 3

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

${(- \sqrt{9 x + 16})}^{2} = {(- x + 4)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 x + 16}$ as ${(9 x + 16)}^{\frac{1}{2}}$ .

${(- {(9 x + 16)}^{\frac{1}{2}})}^{2} = {(- x + 4)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(- {(9 x + 16)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the product rule to $- {(9 x + 16)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(9 x + 16)}^{\frac{1}{2}})}^{2} = {(- x + 4)}^{2}$

Step 4.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(9 x + 16)}^{\frac{1}{2}})}^{2} = {(- x + 4)}^{2}$

Step 4.2.1.3

Multiply ${({(9 x + 16)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(9 x + 16)}^{\frac{1}{2}})}^{2} = {(- x + 4)}^{2}$

Step 4.2.1.4

Multiply the exponents in ${({(9 x + 16)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.4.1

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

${(9 x + 16)}^{\frac{1}{2} \cdot 2} = {(- x + 4)}^{2}$

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Cancel the common factor.

${(9 x + 16)}^{\frac{1}{2} \cdot 2} = {(- x + 4)}^{2}$

Step 4.2.1.4.2.2

Rewrite the expression.

${(9 x + 16)}^{1} = {(- x + 4)}^{2}$

Step 4.2.1.5

Simplify.

$9 x + 16 = {(- x + 4)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(- x + 4)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(- x + 4)}^{2}$ as $(- x + 4) (- x + 4)$ .

$9 x + 16 = (- x + 4) (- x + 4)$

Step 4.3.1.2

Expand $(- x + 4) (- x + 4)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$9 x + 16 = - x (- x + 4) + 4 (- x + 4)$

Step 4.3.1.2.2

Apply the distributive property.

$9 x + 16 = - x (- x) - x \cdot 4 + 4 (- x + 4)$

Step 4.3.1.2.3

Apply the distributive property.

$9 x + 16 = - x (- x) - x \cdot 4 + 4 (- x) + 4 \cdot 4$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$9 x + 16 = - 1 \cdot - 1 x \cdot x - x \cdot 4 + 4 (- x) + 4 \cdot 4$

Step 4.3.1.3.1.2

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

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Step 4.3.1.3.1.2.1

Move $x$ .

$9 x + 16 = - 1 \cdot - 1 (x \cdot x) - x \cdot 4 + 4 (- x) + 4 \cdot 4$

Step 4.3.1.3.1.2.2

Multiply $x$ by $x$ .

$9 x + 16 = - 1 \cdot - 1 x^{2} - x \cdot 4 + 4 (- x) + 4 \cdot 4$

Step 4.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$9 x + 16 = 1 x^{2} - x \cdot 4 + 4 (- x) + 4 \cdot 4$

Step 4.3.1.3.1.4

Multiply $x^{2}$ by $1$ .

$9 x + 16 = x^{2} - x \cdot 4 + 4 (- x) + 4 \cdot 4$

Step 4.3.1.3.1.5

Multiply $4$ by $- 1$ .

$9 x + 16 = x^{2} - 4 x + 4 (- x) + 4 \cdot 4$

Step 4.3.1.3.1.6

Multiply $- 1$ by $4$ .

$9 x + 16 = x^{2} - 4 x - 4 x + 4 \cdot 4$

Step 4.3.1.3.1.7

Multiply $4$ by $4$ .

$9 x + 16 = x^{2} - 4 x - 4 x + 16$

Step 4.3.1.3.2

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

$9 x + 16 = x^{2} - 8 x + 16$

Step 5

Solve for $x$ .

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Step 5.1

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

$x^{2} - 8 x + 16 = 9 x + 16$

Step 5.2

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

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Step 5.2.1

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

$x^{2} - 8 x + 16 - 9 x = 16$

Step 5.2.2

Subtract $9 x$ from $- 8 x$ .

$x^{2} - 17 x + 16 = 16$

Step 5.3

Subtract $16$ from both sides of the equation.

$x^{2} - 17 x + 16 - 16 = 0$

Step 5.4

Combine the opposite terms in $x^{2} - 17 x + 16 - 16$ .

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Step 5.4.1

Subtract $16$ from $16$ .

$x^{2} - 17 x + 0 = 0$

Step 5.4.2

Add $x^{2} - 17 x$ and $0$ .

$x^{2} - 17 x = 0$

Step 5.5

Factor $x$ out of $x^{2} - 17 x$ .

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Step 5.5.1

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

$x \cdot x - 17 x = 0$

Step 5.5.2

Factor $x$ out of $- 17 x$ .

$x \cdot x + x \cdot - 17 = 0$

Step 5.5.3

Factor $x$ out of $x \cdot x + x \cdot - 17$ .

$x (x - 17) = 0$

Step 5.6

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

$x = 0$

$x - 17 = 0$

Step 5.7

Set $x$ equal to $0$ .

$x = 0$

Step 5.8

Set $x - 17$ equal to $0$ and solve for $x$ .

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Step 5.8.1

Set $x - 17$ equal to $0$ .

$x - 17 = 0$

Step 5.8.2

Add $17$ to both sides of the equation.

$x = 17$

Step 5.9

The final solution is all the values that make $x (x - 17) = 0$ true.

$x = 0, 17$

Step 6

Exclude the solutions that do not make $x - 4 - \sqrt{9 x + 16} = 0$ true.

$x = 17$