Precalculus Examples

$\sqrt{3 x} + \sqrt{12} = \frac{x + 7}{\sqrt{3}}$

Step 1

Multiply both sides by $\sqrt{3}$ .

$(\sqrt{3 x} + \sqrt{12}) \sqrt{3} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\sqrt{3 x} + \sqrt{12}) \sqrt{3}$ .

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Step 2.1.1.1

Simplify each term.

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Step 2.1.1.1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.1.1.1.1.1

Factor $4$ out of $12$ .

$(\sqrt{3 x} + \sqrt{4 (3)}) \sqrt{3} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.1.1.2

Rewrite $4$ as $2^{2}$ .

$(\sqrt{3 x} + \sqrt{2^{2} \cdot 3}) \sqrt{3} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.1.2

Pull terms out from under the radical.

$(\sqrt{3 x} + 2 \sqrt{3}) \sqrt{3} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.2

Apply the distributive property.

$\sqrt{3 x} \sqrt{3} + 2 \sqrt{3} \sqrt{3} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.3

Multiply $\sqrt{3 x} \sqrt{3}$ .

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Step 2.1.1.3.1

Combine using the product rule for radicals.

$\sqrt{3 x \cdot 3} + 2 \sqrt{3} \sqrt{3} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.3.2

Multiply $3$ by $3$ .

$\sqrt{9 x} + 2 \sqrt{3} \sqrt{3} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.4

Multiply $2 \sqrt{3} \sqrt{3}$ .

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Step 2.1.1.4.1

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{9 x} + 2 ({\sqrt{3}}^{1} \sqrt{3}) = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{9 x} + 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.4.3

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

$\sqrt{9 x} + 2 {\sqrt{3}}^{1 + 1} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.4.4

Add $1$ and $1$ .

$\sqrt{9 x} + 2 {\sqrt{3}}^{2} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5

Simplify each term.

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Step 2.1.1.5.1

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} x} + 2 {\sqrt{3}}^{2} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.2

Pull terms out from under the radical.

$3 \sqrt{x} + 2 {\sqrt{3}}^{2} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.1.5.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$3 \sqrt{x} + 2 {(3^{\frac{1}{2}})}^{2} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3.2

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

$3 \sqrt{x} + 2 \cdot 3^{\frac{1}{2} \cdot 2} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3.3

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

$3 \sqrt{x} + 2 \cdot 3^{\frac{2}{2}} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3.4

Cancel the common factor of $2$ .

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Step 2.1.1.5.3.4.1

Cancel the common factor.

$3 \sqrt{x} + 2 \cdot 3^{\frac{2}{2}} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3.4.2

Rewrite the expression.

$3 \sqrt{x} + 2 \cdot 3^{1} = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.3.5

Evaluate the exponent.

$3 \sqrt{x} + 2 \cdot 3 = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.1.1.5.4

Multiply $2$ by $3$ .

$3 \sqrt{x} + 6 = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $\sqrt{3}$ .

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Step 2.2.1.1

Cancel the common factor.

$3 \sqrt{x} + 6 = \frac{x + 7}{\sqrt{3}} \sqrt{3}$

Step 2.2.1.2

Rewrite the expression.

$3 \sqrt{x} + 6 = x + 7$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms not containing $3 \sqrt{x}$ to the right side of the equation.

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Step 3.1.1

Subtract $6$ from both sides of the equation.

$3 \sqrt{x} = x + 7 - 6$

Step 3.1.2

Subtract $6$ from $7$ .

$3 \sqrt{x} = x + 1$

Step 3.2

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

${(3 \sqrt{x})}^{2} = {(x + 1)}^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Apply the product rule to $3 x^{\frac{1}{2}}$ .

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

Step 3.3.2.1.2

Raise $3$ to the power of $2$ .

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

Step 3.3.2.1.3

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

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Step 3.3.2.1.3.1

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

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

Step 3.3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.3.2.1

Cancel the common factor.

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

Step 3.3.2.1.3.2.2

Rewrite the expression.

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

Step 3.3.2.1.4

Simplify.

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

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

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Step 3.3.3.1.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$9 x = (x + 1) (x + 1)$

Step 3.3.3.1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 3.3.3.1.2.1

Apply the distributive property.

$9 x = x (x + 1) + 1 (x + 1)$

Step 3.3.3.1.2.2

Apply the distributive property.

$9 x = x \cdot x + x \cdot 1 + 1 (x + 1)$

Step 3.3.3.1.2.3

Apply the distributive property.

$9 x = x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Step 3.3.3.1.3

Simplify and combine like terms.

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Step 3.3.3.1.3.1

Simplify each term.

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Step 3.3.3.1.3.1.1

Multiply $x$ by $x$ .

$9 x = x^{2} + x \cdot 1 + 1 x + 1 \cdot 1$

Step 3.3.3.1.3.1.2

Multiply $x$ by $1$ .

$9 x = x^{2} + x + 1 x + 1 \cdot 1$

Step 3.3.3.1.3.1.3

Multiply $x$ by $1$ .

$9 x = x^{2} + x + x + 1 \cdot 1$

Step 3.3.3.1.3.1.4

Multiply $1$ by $1$ .

$9 x = x^{2} + x + x + 1$

Step 3.3.3.1.3.2

Add $x$ and $x$ .

$9 x = x^{2} + 2 x + 1$

Step 3.4

Solve for $x$ .

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Step 3.4.1

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

$x^{2} + 2 x + 1 = 9 x$

Step 3.4.2

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

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Step 3.4.2.1

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

$x^{2} + 2 x + 1 - 9 x = 0$

Step 3.4.2.2

Subtract $9 x$ from $2 x$ .

$x^{2} - 7 x + 1 = 0$

Step 3.4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4.4

Substitute the values $a = 1$ , $b = - 7$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 3.4.5

Simplify.

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Step 3.4.5.1

Simplify the numerator.

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Step 3.4.5.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.4.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 3.4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 1}}{2 \cdot 1}$

Step 3.4.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 4}}{2 \cdot 1}$

Step 3.4.5.1.3

Subtract $4$ from $49$ .

$x = \frac{7 \pm \sqrt{45}}{2 \cdot 1}$

Step 3.4.5.1.4

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 3.4.5.1.4.1

Factor $9$ out of $45$ .

$x = \frac{7 \pm \sqrt{9 (5)}}{2 \cdot 1}$

Step 3.4.5.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{7 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot 1}$

Step 3.4.5.1.5

Pull terms out from under the radical.

$x = \frac{7 \pm 3 \sqrt{5}}{2 \cdot 1}$

Step 3.4.5.2

Multiply $2$ by $1$ .

$x = \frac{7 \pm 3 \sqrt{5}}{2}$

Step 3.4.6

The final answer is the combination of both solutions.

$x = \frac{7 + 3 \sqrt{5}}{2}, \frac{7 - 3 \sqrt{5}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 + 3 \sqrt{5}}{2}, \frac{7 - 3 \sqrt{5}}{2}$

Decimal Form:

$x = 6.85410196 \dots, 0.14589803 \dots$