Trigonometry Examples

$\sqrt{3 x - 2} + \sqrt{11 + x} = - 1$

Step 1

Subtract $\sqrt{11 + x}$ from both sides of the equation.

$\sqrt{3 x - 2} = - 1 - \sqrt{11 + x}$

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- 1 - \sqrt{11 + x})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(- 1 - \sqrt{11 + x})}^{2}$ as $(- 1 - \sqrt{11 + x}) (- 1 - \sqrt{11 + x})$ .

$3 x - 2 = (- 1 - \sqrt{11 + x}) (- 1 - \sqrt{11 + x})$

Step 3.3.1.2

Expand $(- 1 - \sqrt{11 + x}) (- 1 - \sqrt{11 + x})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$3 x - 2 = - 1 (- 1 - \sqrt{11 + x}) - \sqrt{11 + x} (- 1 - \sqrt{11 + x})$

Step 3.3.1.2.2

Apply the distributive property.

$3 x - 2 = - 1 \cdot - 1 - 1 (- \sqrt{11 + x}) - \sqrt{11 + x} (- 1 - \sqrt{11 + x})$

Step 3.3.1.2.3

Apply the distributive property.

$3 x - 2 = - 1 \cdot - 1 - 1 (- \sqrt{11 + x}) - \sqrt{11 + x} \cdot - 1 - \sqrt{11 + x} (- \sqrt{11 + x})$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 1 - 1 (- \sqrt{11 + x}) - \sqrt{11 + x} \cdot - 1 - \sqrt{11 + x} (- \sqrt{11 + x})$

Step 3.3.1.3.1.2

Multiply $- 1 (- \sqrt{11 + x})$ .

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Step 3.3.1.3.1.2.1

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 1 + 1 \sqrt{11 + x} - \sqrt{11 + x} \cdot - 1 - \sqrt{11 + x} (- \sqrt{11 + x})$

Step 3.3.1.3.1.2.2

Multiply $\sqrt{11 + x}$ by $1$ .

$3 x - 2 = 1 + \sqrt{11 + x} - \sqrt{11 + x} \cdot - 1 - \sqrt{11 + x} (- \sqrt{11 + x})$

Step 3.3.1.3.1.3

Multiply $- \sqrt{11 + x} \cdot - 1$ .

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Step 3.3.1.3.1.3.1

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 1 + \sqrt{11 + x} + 1 \sqrt{11 + x} - \sqrt{11 + x} (- \sqrt{11 + x})$

Step 3.3.1.3.1.3.2

Multiply $\sqrt{11 + x}$ by $1$ .

$3 x - 2 = 1 + \sqrt{11 + x} + \sqrt{11 + x} - \sqrt{11 + x} (- \sqrt{11 + x})$

Step 3.3.1.3.1.4

Multiply $- \sqrt{11 + x} (- \sqrt{11 + x})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 1 + \sqrt{11 + x} + \sqrt{11 + x} + 1 \sqrt{11 + x} \sqrt{11 + x}$

Step 3.3.1.3.1.4.2

Multiply $\sqrt{11 + x}$ by $1$ .

$3 x - 2 = 1 + \sqrt{11 + x} + \sqrt{11 + x} + \sqrt{11 + x} \sqrt{11 + x}$

Step 3.3.1.3.1.4.3

Raise $\sqrt{11 + x}$ to the power of $1$ .

$3 x - 2 = 1 + \sqrt{11 + x} + \sqrt{11 + x} + {\sqrt{11 + x}}^{1} \sqrt{11 + x}$

Step 3.3.1.3.1.4.4

Raise $\sqrt{11 + x}$ to the power of $1$ .

$3 x - 2 = 1 + \sqrt{11 + x} + \sqrt{11 + x} + {\sqrt{11 + x}}^{1} {\sqrt{11 + x}}^{1}$

Step 3.3.1.3.1.4.5

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

$3 x - 2 = 1 + \sqrt{11 + x} + \sqrt{11 + x} + {\sqrt{11 + x}}^{1 + 1}$

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

$3 x - 2 = 1 + \sqrt{11 + x} + \sqrt{11 + x} + {\sqrt{11 + x}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{11 + x}}^{2}$ as $11 + x$ .

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Step 3.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 + x}$ as ${(11 + x)}^{\frac{1}{2}}$ .

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

Step 3.3.1.3.1.5.2

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

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

Step 3.3.1.3.1.5.3

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

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

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

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

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

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

Step 3.3.1.3.1.5.5

Simplify.

$3 x - 2 = 1 + \sqrt{11 + x} + \sqrt{11 + x} + 11 + x$

Step 3.3.1.3.2

Add $1$ and $11$ .

$3 x - 2 = 12 + \sqrt{11 + x} + \sqrt{11 + x} + x$

Step 3.3.1.3.3

Add $\sqrt{11 + x}$ and $\sqrt{11 + x}$ .

$3 x - 2 = 12 + 2 \sqrt{11 + x} + x$

Step 4

Solve for $2 \sqrt{11 + x}$ .

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Step 4.1

Rewrite the equation as $12 + 2 \sqrt{11 + x} + x = 3 x - 2$ .

$12 + 2 \sqrt{11 + x} + x = 3 x - 2$

Step 4.2

Move all terms not containing $2 \sqrt{11 + x}$ to the right side of the equation.

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Step 4.2.1

Subtract $12$ from both sides of the equation.

$2 \sqrt{11 + x} + x = 3 x - 2 - 12$

Step 4.2.2

Subtract $x$ from both sides of the equation.

$2 \sqrt{11 + x} = 3 x - 2 - 12 - x$

Step 4.2.3

Subtract $x$ from $3 x$ .

$2 \sqrt{11 + x} = 2 x - 2 - 12$

Step 4.2.4

Subtract $12$ from $- 2$ .

$2 \sqrt{11 + x} = 2 x - 14$

Step 5

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

${(2 \sqrt{11 + x})}^{2} = {(2 x - 14)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 + x}$ as ${(11 + x)}^{\frac{1}{2}}$ .

${(2 {(11 + x)}^{\frac{1}{2}})}^{2} = {(2 x - 14)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(2 {(11 + x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $2 {(11 + x)}^{\frac{1}{2}}$ .

$2^{2} {({(11 + x)}^{\frac{1}{2}})}^{2} = {(2 x - 14)}^{2}$

Step 6.2.1.2

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

$4 {({(11 + x)}^{\frac{1}{2}})}^{2} = {(2 x - 14)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(11 + x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

$4 {(11 + x)}^{\frac{1}{2} \cdot 2} = {(2 x - 14)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$4 {(11 + x)}^{\frac{1}{2} \cdot 2} = {(2 x - 14)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$4 {(11 + x)}^{1} = {(2 x - 14)}^{2}$

Step 6.2.1.4

Simplify.

$4 (11 + x) = {(2 x - 14)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$4 \cdot 11 + 4 x = {(2 x - 14)}^{2}$

Step 6.2.1.6

Multiply $4$ by $11$ .

$44 + 4 x = {(2 x - 14)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(2 x - 14)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(2 x - 14)}^{2}$ as $(2 x - 14) (2 x - 14)$ .

$44 + 4 x = (2 x - 14) (2 x - 14)$

Step 6.3.1.2

Expand $(2 x - 14) (2 x - 14)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$44 + 4 x = 2 x (2 x - 14) - 14 (2 x - 14)$

Step 6.3.1.2.2

Apply the distributive property.

$44 + 4 x = 2 x (2 x) + 2 x \cdot - 14 - 14 (2 x - 14)$

Step 6.3.1.2.3

Apply the distributive property.

$44 + 4 x = 2 x (2 x) + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$44 + 4 x = 2 \cdot 2 x \cdot x + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Step 6.3.1.3.1.2

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

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Step 6.3.1.3.1.2.1

Move $x$ .

$44 + 4 x = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$44 + 4 x = 2 \cdot 2 x^{2} + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Step 6.3.1.3.1.3

Multiply $2$ by $2$ .

$44 + 4 x = 4 x^{2} + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Step 6.3.1.3.1.4

Multiply $- 14$ by $2$ .

$44 + 4 x = 4 x^{2} - 28 x - 14 (2 x) - 14 \cdot - 14$

Step 6.3.1.3.1.5

Multiply $2$ by $- 14$ .

$44 + 4 x = 4 x^{2} - 28 x - 28 x - 14 \cdot - 14$

Step 6.3.1.3.1.6

Multiply $- 14$ by $- 14$ .

$44 + 4 x = 4 x^{2} - 28 x - 28 x + 196$

Step 6.3.1.3.2

Subtract $28 x$ from $- 28 x$ .

$44 + 4 x = 4 x^{2} - 56 x + 196$

Step 7

Solve for $x$ .

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Step 7.1

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

$4 x^{2} - 56 x + 196 = 44 + 4 x$

Step 7.2

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

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Step 7.2.1

Subtract $4 x$ from both sides of the equation.

$4 x^{2} - 56 x + 196 - 4 x = 44$

Step 7.2.2

Subtract $4 x$ from $- 56 x$ .

$4 x^{2} - 60 x + 196 = 44$

Step 7.3

Subtract $44$ from both sides of the equation.

$4 x^{2} - 60 x + 196 - 44 = 0$

Step 7.4

Subtract $44$ from $196$ .

$4 x^{2} - 60 x + 152 = 0$

Step 7.5

Factor $4$ out of $4 x^{2} - 60 x + 152$ .

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Step 7.5.1

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) - 60 x + 152 = 0$

Step 7.5.2

Factor $4$ out of $- 60 x$ .

$4 (x^{2}) + 4 (- 15 x) + 152 = 0$

Step 7.5.3

Factor $4$ out of $152$ .

$4 x^{2} + 4 (- 15 x) + 4 \cdot 38 = 0$

Step 7.5.4

Factor $4$ out of $4 x^{2} + 4 (- 15 x)$ .

$4 (x^{2} - 15 x) + 4 \cdot 38 = 0$

Step 7.5.5

Factor $4$ out of $4 (x^{2} - 15 x) + 4 \cdot 38$ .

$4 (x^{2} - 15 x + 38) = 0$

Step 7.6

Divide each term in $4 (x^{2} - 15 x + 38) = 0$ by $4$ and simplify.

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Step 7.6.1

Divide each term in $4 (x^{2} - 15 x + 38) = 0$ by $4$ .

$\frac{4 (x^{2} - 15 x + 38)}{4} = \frac{0}{4}$

Step 7.6.2

Simplify the left side.

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Step 7.6.2.1

Cancel the common factor of $4$ .

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Step 7.6.2.1.1

Cancel the common factor.

$\frac{4 (x^{2} - 15 x + 38)}{4} = \frac{0}{4}$

Step 7.6.2.1.2

Divide $x^{2} - 15 x + 38$ by $1$ .

$x^{2} - 15 x + 38 = \frac{0}{4}$

Step 7.6.3

Simplify the right side.

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Step 7.6.3.1

Divide $0$ by $4$ .

$x^{2} - 15 x + 38 = 0$

Step 7.7

Use the quadratic formula to find the solutions.

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

Step 7.8

Substitute the values $a = 1$ , $b = - 15$ , and $c = 38$ into the quadratic formula and solve for $x$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (1 \cdot 38)}}{2 \cdot 1}$

Step 7.9

Simplify.

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Step 7.9.1

Simplify the numerator.

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Step 7.9.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot 38}}{2 \cdot 1}$

Step 7.9.1.2

Multiply $- 4 \cdot 1 \cdot 38$ .

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Step 7.9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 38}}{2 \cdot 1}$

Step 7.9.1.2.2

Multiply $- 4$ by $38$ .

$x = \frac{15 \pm \sqrt{225 - 152}}{2 \cdot 1}$

Step 7.9.1.3

Subtract $152$ from $225$ .

$x = \frac{15 \pm \sqrt{73}}{2 \cdot 1}$

Step 7.9.2

Multiply $2$ by $1$ .

$x = \frac{15 \pm \sqrt{73}}{2}$

Step 7.10

The final answer is the combination of both solutions.

$x = \frac{15 + \sqrt{73}}{2}, \frac{15 - \sqrt{73}}{2}$

Step 8

Exclude the solutions that do not make $\sqrt{3 x - 2} + \sqrt{11 + x} = - 1$ true.

$No solution$