Finite Math Examples

$\sqrt{9 x + 27} = 2 x + 7$

Step 1

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

${\sqrt{9 x + 27}}^{2} = {(2 x + 7)}^{2}$

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 $\sqrt{9 x + 27}$ as ${(9 x + 27)}^{\frac{1}{2}}$ .

${({(9 x + 27)}^{\frac{1}{2}})}^{2} = {(2 x + 7)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(9 x + 27)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(9 x + 27)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(9 x + 27)}^{\frac{1}{2} \cdot 2} = {(2 x + 7)}^{2}$

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.

${(9 x + 27)}^{\frac{1}{2} \cdot 2} = {(2 x + 7)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(9 x + 27)}^{1} = {(2 x + 7)}^{2}$

Step 2.2.1.2

Simplify.

$9 x + 27 = {(2 x + 7)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(2 x + 7)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(2 x + 7)}^{2}$ as $(2 x + 7) (2 x + 7)$ .

$9 x + 27 = (2 x + 7) (2 x + 7)$

Step 2.3.1.2

Expand $(2 x + 7) (2 x + 7)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$9 x + 27 = 2 x (2 x + 7) + 7 (2 x + 7)$

Step 2.3.1.2.2

Apply the distributive property.

$9 x + 27 = 2 x (2 x) + 2 x \cdot 7 + 7 (2 x + 7)$

Step 2.3.1.2.3

Apply the distributive property.

$9 x + 27 = 2 x (2 x) + 2 x \cdot 7 + 7 (2 x) + 7 \cdot 7$

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

Rewrite using the commutative property of multiplication.

$9 x + 27 = 2 \cdot 2 x \cdot x + 2 x \cdot 7 + 7 (2 x) + 7 \cdot 7$

Step 2.3.1.3.1.2

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

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Step 2.3.1.3.1.2.1

Move $x$ .

$9 x + 27 = 2 \cdot 2 (x \cdot x) + 2 x \cdot 7 + 7 (2 x) + 7 \cdot 7$

Step 2.3.1.3.1.2.2

Multiply $x$ by $x$ .

$9 x + 27 = 2 \cdot 2 x^{2} + 2 x \cdot 7 + 7 (2 x) + 7 \cdot 7$

Step 2.3.1.3.1.3

Multiply $2$ by $2$ .

$9 x + 27 = 4 x^{2} + 2 x \cdot 7 + 7 (2 x) + 7 \cdot 7$

Step 2.3.1.3.1.4

Multiply $7$ by $2$ .

$9 x + 27 = 4 x^{2} + 14 x + 7 (2 x) + 7 \cdot 7$

Step 2.3.1.3.1.5

Multiply $2$ by $7$ .

$9 x + 27 = 4 x^{2} + 14 x + 14 x + 7 \cdot 7$

Step 2.3.1.3.1.6

Multiply $7$ by $7$ .

$9 x + 27 = 4 x^{2} + 14 x + 14 x + 49$

Step 2.3.1.3.2

Add $14 x$ and $14 x$ .

$9 x + 27 = 4 x^{2} + 28 x + 49$

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.

$4 x^{2} + 28 x + 49 = 9 x + 27$

Step 3.2

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

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Step 3.2.1

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

$4 x^{2} + 28 x + 49 - 9 x = 27$

Step 3.2.2

Subtract $9 x$ from $28 x$ .

$4 x^{2} + 19 x + 49 = 27$

Step 3.3

Subtract $27$ from both sides of the equation.

$4 x^{2} + 19 x + 49 - 27 = 0$

Step 3.4

Subtract $27$ from $49$ .

$4 x^{2} + 19 x + 22 = 0$

Step 3.5

Factor by grouping.

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Step 3.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 22 = 88$ and whose sum is $b = 19$ .

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Step 3.5.1.1

Factor $19$ out of $19 x$ .

$4 x^{2} + 19 (x) + 22 = 0$

Step 3.5.1.2

Rewrite $19$ as $8$ plus $11$

$4 x^{2} + (8 + 11) x + 22 = 0$

Step 3.5.1.3

Apply the distributive property.

$4 x^{2} + 8 x + 11 x + 22 = 0$

Step 3.5.2

Factor out the greatest common factor from each group.

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Step 3.5.2.1

Group the first two terms and the last two terms.

$(4 x^{2} + 8 x) + 11 x + 22 = 0$

Step 3.5.2.2

Factor out the greatest common factor (GCF) from each group.

$4 x (x + 2) + 11 (x + 2) = 0$

Step 3.5.3

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

$(x + 2) (4 x + 11) = 0$

Step 3.6

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

$x + 2 = 0$

$4 x + 11 = 0$

Step 3.7

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.8

Set $4 x + 11$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $4 x + 11$ equal to $0$ .

$4 x + 11 = 0$

Step 3.8.2

Solve $4 x + 11 = 0$ for $x$ .

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Step 3.8.2.1

Subtract $11$ from both sides of the equation.

$4 x = - 11$

Step 3.8.2.2

Divide each term in $4 x = - 11$ by $4$ and simplify.

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Step 3.8.2.2.1

Divide each term in $4 x = - 11$ by $4$ .

$\frac{4 x}{4} = \frac{- 11}{4}$

Step 3.8.2.2.2

Simplify the left side.

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Step 3.8.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.8.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 11}{4}$

Step 3.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{4}$

Step 3.8.2.2.3

Simplify the right side.

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Step 3.8.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{11}{4}$

Step 3.9

The final solution is all the values that make $(x + 2) (4 x + 11) = 0$ true.

$x = - 2, - \frac{11}{4}$

Step 4