Finite Math Examples

$\sqrt{18 x - 27} = x + 3$

Step 1

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

${\sqrt{18 x - 27}}^{2} = {(x + 3)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{18 x - 27}$ as ${(18 x - 27)}^{\frac{1}{2}}$ .

${({(18 x - 27)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(18 x - 27)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(18 x - 27)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(18 x - 27)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(18 x - 27)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(18 x - 27)}^{1} = {(x + 3)}^{2}$

Step 2.2.1.2

Simplify.

$18 x - 27 = {(x + 3)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(x + 3)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$18 x - 27 = (x + 3) (x + 3)$

Step 2.3.1.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$18 x - 27 = x (x + 3) + 3 (x + 3)$

Step 2.3.1.2.2

Apply the distributive property.

$18 x - 27 = x \cdot x + x \cdot 3 + 3 (x + 3)$

Step 2.3.1.2.3

Apply the distributive property.

$18 x - 27 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $x$ by $x$ .

$18 x - 27 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Step 2.3.1.3.1.2

Move $3$ to the left of $x$ .

$18 x - 27 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Step 2.3.1.3.1.3

Multiply $3$ by $3$ .

$18 x - 27 = x^{2} + 3 x + 3 x + 9$

Step 2.3.1.3.2

Add $3 x$ and $3 x$ .

$18 x - 27 = x^{2} + 6 x + 9$

Step 3

Solve for $x$ .

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Step 3.1

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

$x^{2} + 6 x + 9 = 18 x - 27$

Step 3.2

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

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Step 3.2.1

Subtract $18 x$ from both sides of the equation.

$x^{2} + 6 x + 9 - 18 x = - 27$

Step 3.2.2

Subtract $18 x$ from $6 x$ .

$x^{2} - 12 x + 9 = - 27$

Step 3.3

Add $27$ to both sides of the equation.

$x^{2} - 12 x + 9 + 27 = 0$

Step 3.4

Add $9$ and $27$ .

$x^{2} - 12 x + 36 = 0$

Step 3.5

Factor using the perfect square rule.

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Step 3.5.1

Rewrite $36$ as $6^{2}$ .

$x^{2} - 12 x + 6^{2} = 0$

Step 3.5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 x = 2 \cdot x \cdot 6$

Step 3.5.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 6 + 6^{2} = 0$

Step 3.5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

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

Step 3.6

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.7

Add $6$ to both sides of the equation.

$x = 6$