Trigonometry Examples

$\sqrt{3 x + 7} = x - 71$

Step 1

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

${\sqrt{3 x + 7}}^{2} = {(x - 71)}^{2}$

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

${({(3 x + 7)}^{\frac{1}{2}})}^{2} = {(x - 71)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(3 x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(3 x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(3 x + 7)}^{\frac{1}{2} \cdot 2} = {(x - 71)}^{2}$

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.

${(3 x + 7)}^{\frac{1}{2} \cdot 2} = {(x - 71)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(3 x + 7)}^{1} = {(x - 71)}^{2}$

Step 2.2.1.2

Simplify.

$3 x + 7 = {(x - 71)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(x - 71)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(x - 71)}^{2}$ as $(x - 71) (x - 71)$ .

$3 x + 7 = (x - 71) (x - 71)$

Step 2.3.1.2

Expand $(x - 71) (x - 71)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$3 x + 7 = x (x - 71) - 71 (x - 71)$

Step 2.3.1.2.2

Apply the distributive property.

$3 x + 7 = x \cdot x + x \cdot - 71 - 71 (x - 71)$

Step 2.3.1.2.3

Apply the distributive property.

$3 x + 7 = x \cdot x + x \cdot - 71 - 71 x - 71 \cdot - 71$

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 $x$ .

$3 x + 7 = x^{2} + x \cdot - 71 - 71 x - 71 \cdot - 71$

Step 2.3.1.3.1.2

Move $- 71$ to the left of $x$ .

$3 x + 7 = x^{2} - 71 \cdot x - 71 x - 71 \cdot - 71$

Step 2.3.1.3.1.3

Multiply $- 71$ by $- 71$ .

$3 x + 7 = x^{2} - 71 x - 71 x + 5041$

Step 2.3.1.3.2

Subtract $71 x$ from $- 71 x$ .

$3 x + 7 = x^{2} - 142 x + 5041$

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.

$x^{2} - 142 x + 5041 = 3 x + 7$

Step 3.2

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

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Step 3.2.1

Subtract $3 x$ from both sides of the equation.

$x^{2} - 142 x + 5041 - 3 x = 7$

Step 3.2.2

Subtract $3 x$ from $- 142 x$ .

$x^{2} - 145 x + 5041 = 7$

Step 3.3

Move all terms to the left side of the equation and simplify.

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Step 3.3.1

Subtract $7$ from both sides of the equation.

$x^{2} - 145 x + 5041 - 7 = 0$

Step 3.3.2

Subtract $7$ from $5041$ .

$x^{2} - 145 x + 5034 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 1$ , $b = - 145$ , and $c = 5034$ into the quadratic formula and solve for $x$ .

$\frac{145 \pm \sqrt{{(- 145)}^{2} - 4 \cdot (1 \cdot 5034)}}{2 \cdot 1}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 145$ to the power of $2$ .

$x = \frac{145 \pm \sqrt{21025 - 4 \cdot 1 \cdot 5034}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 4 \cdot 1 \cdot 5034$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{145 \pm \sqrt{21025 - 4 \cdot 5034}}{2 \cdot 1}$

Step 3.6.1.2.2

Multiply $- 4$ by $5034$ .

$x = \frac{145 \pm \sqrt{21025 - 20136}}{2 \cdot 1}$

Step 3.6.1.3

Subtract $20136$ from $21025$ .

$x = \frac{145 \pm \sqrt{889}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$x = \frac{145 \pm \sqrt{889}}{2}$

Step 3.7

The final answer is the combination of both solutions.

$x = \frac{145 + \sqrt{889}}{2}, \frac{145 - \sqrt{889}}{2}$

Step 4

Exclude the solutions that do not make $\sqrt{3 x + 7} = x - 71$ true.

$x = \frac{145 + \sqrt{889}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{145 + \sqrt{889}}{2}$

Decimal Form:

$x = 87.40805151 \dots$