Algebra Examples

$\sqrt{x + 2} + \sqrt{3 x + 10} = 2$

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{x + 2} + \sqrt{3 x + 10} - 2 = 0$

Step 2

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

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Step 2.1

Subtract $\sqrt{3 x + 10}$ from both sides of the equation.

$\sqrt{x + 2} - 2 = - \sqrt{3 x + 10}$

Step 2.2

Add $2$ to both sides of the equation.

$\sqrt{x + 2} = - \sqrt{3 x + 10} + 2$

Step 3

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

${\sqrt{x + 2}}^{2} = {(- \sqrt{3 x + 10} + 2)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

${({(x + 2)}^{\frac{1}{2}})}^{2} = {(- \sqrt{3 x + 10} + 2)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

${(x + 2)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{3 x + 10} + 2)}^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(x + 2)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{3 x + 10} + 2)}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(x + 2)}^{1} = {(- \sqrt{3 x + 10} + 2)}^{2}$

Step 4.2.1.2

Simplify.

$x + 2 = {(- \sqrt{3 x + 10} + 2)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(- \sqrt{3 x + 10} + 2)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(- \sqrt{3 x + 10} + 2)}^{2}$ as $(- \sqrt{3 x + 10} + 2) (- \sqrt{3 x + 10} + 2)$ .

$x + 2 = (- \sqrt{3 x + 10} + 2) (- \sqrt{3 x + 10} + 2)$

Step 4.3.1.2

Expand $(- \sqrt{3 x + 10} + 2) (- \sqrt{3 x + 10} + 2)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$x + 2 = - \sqrt{3 x + 10} (- \sqrt{3 x + 10} + 2) + 2 (- \sqrt{3 x + 10} + 2)$

Step 4.3.1.2.2

Apply the distributive property.

$x + 2 = - \sqrt{3 x + 10} (- \sqrt{3 x + 10}) - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10} + 2)$

Step 4.3.1.2.3

Apply the distributive property.

$x + 2 = - \sqrt{3 x + 10} (- \sqrt{3 x + 10}) - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Multiply $- \sqrt{3 x + 10} (- \sqrt{3 x + 10})$ .

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Step 4.3.1.3.1.1.1

Multiply $- 1$ by $- 1$ .

$x + 2 = 1 \sqrt{3 x + 10} \sqrt{3 x + 10} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.1.2

Multiply $\sqrt{3 x + 10}$ by $1$ .

$x + 2 = \sqrt{3 x + 10} \sqrt{3 x + 10} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.1.3

Raise $\sqrt{3 x + 10}$ to the power of $1$ .

$x + 2 = {\sqrt{3 x + 10}}^{1} \sqrt{3 x + 10} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.1.4

Raise $\sqrt{3 x + 10}$ to the power of $1$ .

$x + 2 = {\sqrt{3 x + 10}}^{1} {\sqrt{3 x + 10}}^{1} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.1.5

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

$x + 2 = {\sqrt{3 x + 10}}^{1 + 1} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.1.6

Add $1$ and $1$ .

$x + 2 = {\sqrt{3 x + 10}}^{2} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.2

Rewrite ${\sqrt{3 x + 10}}^{2}$ as $3 x + 10$ .

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Step 4.3.1.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 10}$ as ${(3 x + 10)}^{\frac{1}{2}}$ .

$x + 2 = {({(3 x + 10)}^{\frac{1}{2}})}^{2} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.2.2

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

$x + 2 = {(3 x + 10)}^{\frac{1}{2} \cdot 2} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.2.3

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

$x + 2 = {(3 x + 10)}^{\frac{2}{2}} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 4.3.1.3.1.2.4.1

Cancel the common factor.

$x + 2 = {(3 x + 10)}^{\frac{2}{2}} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.2.4.2

Rewrite the expression.

$x + 2 = {(3 x + 10)}^{1} - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.2.5

Simplify.

$x + 2 = 3 x + 10 - \sqrt{3 x + 10} \cdot 2 + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.3

Multiply $2$ by $- 1$ .

$x + 2 = 3 x + 10 - 2 \sqrt{3 x + 10} + 2 (- \sqrt{3 x + 10}) + 2 \cdot 2$

Step 4.3.1.3.1.4

Multiply $- 1$ by $2$ .

$x + 2 = 3 x + 10 - 2 \sqrt{3 x + 10} - 2 \sqrt{3 x + 10} + 2 \cdot 2$

Step 4.3.1.3.1.5

Multiply $2$ by $2$ .

$x + 2 = 3 x + 10 - 2 \sqrt{3 x + 10} - 2 \sqrt{3 x + 10} + 4$

Step 4.3.1.3.2

Add $10$ and $4$ .

$x + 2 = 3 x + 14 - 2 \sqrt{3 x + 10} - 2 \sqrt{3 x + 10}$

Step 4.3.1.3.3

Subtract $2 \sqrt{3 x + 10}$ from $- 2 \sqrt{3 x + 10}$ .

$x + 2 = 3 x + 14 - 4 \sqrt{3 x + 10}$

Step 5

Solve for $- 4 \sqrt{3 x + 10}$ .

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Step 5.1

Rewrite the equation as $3 x + 14 - 4 \sqrt{3 x + 10} = x + 2$ .

$3 x + 14 - 4 \sqrt{3 x + 10} = x + 2$

Step 5.2

Move all terms not containing $- 4 \sqrt{3 x + 10}$ to the right side of the equation.

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Step 5.2.1

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

$14 - 4 \sqrt{3 x + 10} = x + 2 - 3 x$

Step 5.2.2

Subtract $14$ from both sides of the equation.

$- 4 \sqrt{3 x + 10} = x + 2 - 3 x - 14$

Step 5.2.3

Subtract $3 x$ from $x$ .

$- 4 \sqrt{3 x + 10} = - 2 x + 2 - 14$

Step 5.2.4

Subtract $14$ from $2$ .

$- 4 \sqrt{3 x + 10} = - 2 x - 12$

Step 6

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

${(- 4 \sqrt{3 x + 10})}^{2} = {(- 2 x - 12)}^{2}$

Step 7

Simplify each side of the equation.

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Step 7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 10}$ as ${(3 x + 10)}^{\frac{1}{2}}$ .

${(- 4 {(3 x + 10)}^{\frac{1}{2}})}^{2} = {(- 2 x - 12)}^{2}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify ${(- 4 {(3 x + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 7.2.1.1

Apply the product rule to $- 4 {(3 x + 10)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(3 x + 10)}^{\frac{1}{2}})}^{2} = {(- 2 x - 12)}^{2}$

Step 7.2.1.2

Raise $- 4$ to the power of $2$ .

$16 {({(3 x + 10)}^{\frac{1}{2}})}^{2} = {(- 2 x - 12)}^{2}$

Step 7.2.1.3

Multiply the exponents in ${({(3 x + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 7.2.1.3.1

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

$16 {(3 x + 10)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 12)}^{2}$

Step 7.2.1.3.2

Cancel the common factor of $2$ .

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Step 7.2.1.3.2.1

Cancel the common factor.

$16 {(3 x + 10)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 12)}^{2}$

Step 7.2.1.3.2.2

Rewrite the expression.

$16 {(3 x + 10)}^{1} = {(- 2 x - 12)}^{2}$

Step 7.2.1.4

Simplify.

$16 (3 x + 10) = {(- 2 x - 12)}^{2}$

Step 7.2.1.5

Apply the distributive property.

$16 (3 x) + 16 \cdot 10 = {(- 2 x - 12)}^{2}$

Step 7.2.1.6

Multiply.

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Step 7.2.1.6.1

Multiply $3$ by $16$ .

$48 x + 16 \cdot 10 = {(- 2 x - 12)}^{2}$

Step 7.2.1.6.2

Multiply $16$ by $10$ .

$48 x + 160 = {(- 2 x - 12)}^{2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify ${(- 2 x - 12)}^{2}$ .

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Step 7.3.1.1

Rewrite ${(- 2 x - 12)}^{2}$ as $(- 2 x - 12) (- 2 x - 12)$ .

$48 x + 160 = (- 2 x - 12) (- 2 x - 12)$

Step 7.3.1.2

Expand $(- 2 x - 12) (- 2 x - 12)$ using the FOIL Method.

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Step 7.3.1.2.1

Apply the distributive property.

$48 x + 160 = - 2 x (- 2 x - 12) - 12 (- 2 x - 12)$

Step 7.3.1.2.2

Apply the distributive property.

$48 x + 160 = - 2 x (- 2 x) - 2 x \cdot - 12 - 12 (- 2 x - 12)$

Step 7.3.1.2.3

Apply the distributive property.

$48 x + 160 = - 2 x (- 2 x) - 2 x \cdot - 12 - 12 (- 2 x) - 12 \cdot - 12$

Step 7.3.1.3

Simplify and combine like terms.

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Step 7.3.1.3.1

Simplify each term.

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Step 7.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$48 x + 160 = - 2 \cdot - 2 x \cdot x - 2 x \cdot - 12 - 12 (- 2 x) - 12 \cdot - 12$

Step 7.3.1.3.1.2

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

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Step 7.3.1.3.1.2.1

Move $x$ .

$48 x + 160 = - 2 \cdot - 2 (x \cdot x) - 2 x \cdot - 12 - 12 (- 2 x) - 12 \cdot - 12$

Step 7.3.1.3.1.2.2

Multiply $x$ by $x$ .

$48 x + 160 = - 2 \cdot - 2 x^{2} - 2 x \cdot - 12 - 12 (- 2 x) - 12 \cdot - 12$

Step 7.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

$48 x + 160 = 4 x^{2} - 2 x \cdot - 12 - 12 (- 2 x) - 12 \cdot - 12$

Step 7.3.1.3.1.4

Multiply $- 12$ by $- 2$ .

$48 x + 160 = 4 x^{2} + 24 x - 12 (- 2 x) - 12 \cdot - 12$

Step 7.3.1.3.1.5

Multiply $- 2$ by $- 12$ .

$48 x + 160 = 4 x^{2} + 24 x + 24 x - 12 \cdot - 12$

Step 7.3.1.3.1.6

Multiply $- 12$ by $- 12$ .

$48 x + 160 = 4 x^{2} + 24 x + 24 x + 144$

Step 7.3.1.3.2

Add $24 x$ and $24 x$ .

$48 x + 160 = 4 x^{2} + 48 x + 144$

Step 8

Solve for $x$ .

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Step 8.1

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

$4 x^{2} + 48 x + 144 = 48 x + 160$

Step 8.2

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

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Step 8.2.1

Subtract $48 x$ from both sides of the equation.

$4 x^{2} + 48 x + 144 - 48 x = 160$

Step 8.2.2

Combine the opposite terms in $4 x^{2} + 48 x + 144 - 48 x$ .

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Step 8.2.2.1

Subtract $48 x$ from $48 x$ .

$4 x^{2} + 0 + 144 = 160$

Step 8.2.2.2

Add $4 x^{2}$ and $0$ .

$4 x^{2} + 144 = 160$

Step 8.3

Move all terms not containing $x$ to the right side of the equation.

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Step 8.3.1

Subtract $144$ from both sides of the equation.

$4 x^{2} = 160 - 144$

Step 8.3.2

Subtract $144$ from $160$ .

$4 x^{2} = 16$

Step 8.4

Divide each term in $4 x^{2} = 16$ by $4$ and simplify.

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Step 8.4.1

Divide each term in $4 x^{2} = 16$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{16}{4}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Cancel the common factor of $4$ .

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Step 8.4.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{16}{4}$

Step 8.4.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{16}{4}$

Step 8.4.3

Simplify the right side.

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Step 8.4.3.1

Divide $16$ by $4$ .

$x^{2} = 4$

Step 8.5

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

$x = \pm \sqrt{4}$

Step 8.6

Simplify $\pm \sqrt{4}$ .

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Step 8.6.1

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

$x = \pm \sqrt{2^{2}}$

Step 8.6.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 8.7

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

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Step 8.7.1

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

$x = 2$

Step 8.7.2

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

$x = - 2$

Step 8.7.3

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

$x = 2, - 2$

Step 9

Exclude the solutions that do not make $\sqrt{x + 2} + \sqrt{3 x + 10} - 2 = 0$ true.

$x = - 2$