Finite Math Examples

$3 {(x + 3)}^{2} = 11$

Step 1

Divide each term in $3 {(x + 3)}^{2} = 11$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 {(x + 3)}^{2} = 11$ by $3$ .

$\frac{3 {(x + 3)}^{2}}{3} = \frac{11}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 {(x + 3)}^{2}}{3} = \frac{11}{3}$

Step 1.2.1.2

Divide ${(x + 3)}^{2}$ by $1$ .

${(x + 3)}^{2} = \frac{11}{3}$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + 3 = \pm \sqrt{\frac{11}{3}}$

Step 3

Simplify $\pm \sqrt{\frac{11}{3}}$ .

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Step 3.1

Rewrite $\sqrt{\frac{11}{3}}$ as $\frac{\sqrt{11}}{\sqrt{3}}$ .

$x + 3 = \pm \frac{\sqrt{11}}{\sqrt{3}}$

Step 3.2

Multiply $\frac{\sqrt{11}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x + 3 = \pm \frac{\sqrt{11}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{\sqrt{11}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.3.2

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

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.3.3

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

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.3.4

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

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.3.5

Add $1$ and $1$ .

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.3.6

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

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Step 3.3.6.1

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

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.3.6.2

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

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.3.6.3

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

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.4.1

Cancel the common factor.

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.3.6.4.2

Rewrite the expression.

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{3^{1}}$

Step 3.3.6.5

Evaluate the exponent.

$x + 3 = \pm \frac{\sqrt{11} \sqrt{3}}{3}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Combine using the product rule for radicals.

$x + 3 = \pm \frac{\sqrt{11 \cdot 3}}{3}$

Step 3.4.2

Multiply $11$ by $3$ .

$x + 3 = \pm \frac{\sqrt{33}}{3}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$x + 3 = \frac{\sqrt{33}}{3}$

Step 4.2

Subtract $3$ from both sides of the equation.

$x = \frac{\sqrt{33}}{3} - 3$

Step 4.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + 3 = - \frac{\sqrt{33}}{3}$

Step 4.4

Subtract $3$ from both sides of the equation.

$x = - \frac{\sqrt{33}}{3} - 3$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{33}}{3} - 3, - \frac{\sqrt{33}}{3} - 3$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{33}}{3} - 3, - \frac{\sqrt{33}}{3} - 3$

Decimal Form:

$x = - 1.08514578 \dots, - 4.91485421 \dots$