Finite Math Examples

\sqrt{16 - 6 x} - x = 0

$\sqrt{16 - 6 x} - x = 0$

Step 1

Factor

2

$2$ out of

16 - 6 x

$16 - 6 x$ .

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Step 1.1

Factor

2

$2$ out of

16

$16$ .

\sqrt{2 (8) - 6 x} - x = 0

$\sqrt{2 (8) - 6 x} - x = 0$

Step 1.2

Factor

2

$2$ out of

- 6 x

$- 6 x$ .

\sqrt{2 (8) + 2 (- 3 x)} - x = 0

$\sqrt{2 (8) + 2 (- 3 x)} - x = 0$

Step 1.3

Factor

2

$2$ out of

2 (8) + 2 (- 3 x)

$2 (8) + 2 (- 3 x)$ .

\sqrt{2 (8 - 3 x)} - x = 0

$\sqrt{2 (8 - 3 x)} - x = 0$

\sqrt{2 (8 - 3 x)} - x = 0

$\sqrt{2 (8 - 3 x)} - x = 0$

Step 2

Add

x

$x$ to both sides of the equation.

\sqrt{2 (8 - 3 x)} = x

$\sqrt{2 (8 - 3 x)} = x$

Step 3

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

{\sqrt{2 (8 - 3 x)}}^{2} = x^{2}

${\sqrt{2 (8 - 3 x)}}^{2} = x^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2 (8 - 3 x)}

$\sqrt{2 (8 - 3 x)}$ as

{(2 (8 - 3 x))}^{\frac{1}{2}}

${(2 (8 - 3 x))}^{\frac{1}{2}}$ .

{({(2 (8 - 3 x))}^{\frac{1}{2}})}^{2} = x^{2}

${({(2 (8 - 3 x))}^{\frac{1}{2}})}^{2} = x^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify

{({(2 (8 - 3 x))}^{\frac{1}{2}})}^{2}

${({(2 (8 - 3 x))}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in

{({(2 (8 - 3 x))}^{\frac{1}{2}})}^{2}

${({(2 (8 - 3 x))}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

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

{(2 (8 - 3 x))}^{\frac{1}{2} \cdot 2} = x^{2}

${(2 (8 - 3 x))}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 4.2.1.1.2

Cancel the common factor of

2

$2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(2 (8 - 3 x))}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(2 (8 - 3 x))}^{1} = x^{2}$

Step 4.2.1.2

Apply the distributive property.

${(2 \cdot 8 + 2 (- 3 x))}^{1} = x^{2}$

Step 4.2.1.3

Multiply.

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Step 4.2.1.3.1

Multiply

$2$ by

$8$ .

${(16 + 2 (- 3 x))}^{1} = x^{2}$

Step 4.2.1.3.2

Multiply

$- 3$ by

$2$ .

${(16 - 6 x)}^{1} = x^{2}$

Step 4.2.1.3.3

Simplify.

$16 - 6 x = x^{2}$

Step 5

Solve for

$x$ .

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Step 5.1

Subtract

$x^{2}$ from both sides of the equation.

$16 - 6 x - x^{2} = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Factor

$- 1$ out of

$16 - 6 x - x^{2}$ .

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Step 5.2.1.1

Reorder the expression.

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Step 5.2.1.1.1

Move

$16$ .

$- 6 x - x^{2} + 16 = 0$

Step 5.2.1.1.2

Reorder

$- 6 x$ and

$- x^{2}$ .

$- x^{2} - 6 x + 16 = 0$

Step 5.2.1.2

Factor

$- 1$ out of

$- x^{2}$ .

$- (x^{2}) - 6 x + 16 = 0$

Step 5.2.1.3

Factor

$- 1$ out of

$- 6 x$ .

$- (x^{2}) - (6 x) + 16 = 0$

Step 5.2.1.4

Rewrite

$16$ as

$- 1 (- 16)$ .

$- (x^{2}) - (6 x) - 1 \cdot - 16 = 0$

Step 5.2.1.5

Factor

$- 1$ out of

$- (x^{2}) - (6 x)$ .

$- (x^{2} + 6 x) - 1 \cdot - 16 = 0$

Step 5.2.1.6

Factor

$- 1$ out of

$- (x^{2} + 6 x) - 1 (- 16)$ .

$- (x^{2} + 6 x - 16) = 0$

Step 5.2.2

Factor.

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Step 5.2.2.1

Factor

$x^{2} + 6 x - 16$ using the AC method.

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Step 5.2.2.1.1

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

$- 16$ and whose sum is

$6$ .

$- 2, 8$

Step 5.2.2.1.2

Write the factored form using these integers.

$- ((x - 2) (x + 8)) = 0$

Step 5.2.2.2

Remove unnecessary parentheses.

$- (x - 2) (x + 8) = 0$

Step 5.3

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x - 2 = 0$

$x + 8 = 0$

Step 5.4

Set

$x - 2$ equal to

$0$ and solve for

$x$ .

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Step 5.4.1

Set

$x - 2$ equal to

$0$ .

$x - 2 = 0$

Step 5.4.2

Add

$2$ to both sides of the equation.

$x = 2$

Step 5.5

Set

$x + 8$ equal to

$0$ and solve for

$x$ .

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Step 5.5.1

Set

$x + 8$ equal to

$0$ .

$x + 8 = 0$

Step 5.5.2

Subtract

$8$ from both sides of the equation.

$x = - 8$

Step 5.6

The final solution is all the values that make

$- (x - 2) (x + 8) = 0$ true.

$x = 2, - 8$

Step 6

Exclude the solutions that do not make

$\sqrt{2 (8 - 3 x)} - x = 0$ true.

$x = 2$