Algebra Examples

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

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

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

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

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite ${(\sqrt{3 - x} - 1)}^{2}$ as $(\sqrt{3 - x} - 1) (\sqrt{3 - x} - 1)$ .

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

Step 3.3.1.2

Expand $(\sqrt{3 - x} - 1) (\sqrt{3 - x} - 1)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

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

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

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Step 3.3.1.3.1.1.1

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

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

Step 3.3.1.3.1.1.2

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

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

Step 3.3.1.3.1.1.3

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

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

Step 3.3.1.3.1.1.4

Add $1$ and $1$ .

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

Step 3.3.1.3.1.2

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

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Step 3.3.1.3.1.2.1

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

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

Step 3.3.1.3.1.2.2

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

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

Step 3.3.1.3.1.2.3

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

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

Step 3.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.2.4.1

Cancel the common factor.

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

Step 3.3.1.3.1.2.4.2

Rewrite the expression.

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

Step 3.3.1.3.1.2.5

Simplify.

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

Step 3.3.1.3.1.3

Move $- 1$ to the left of $\sqrt{3 - x}$ .

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

Step 3.3.1.3.1.4

Rewrite $- 1 \sqrt{3 - x}$ as $- \sqrt{3 - x}$ .

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

Step 3.3.1.3.1.5

Rewrite $- 1 \sqrt{3 - x}$ as $- \sqrt{3 - x}$ .

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

Step 3.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.3.2

Add $3$ and $1$ .

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

Step 3.3.1.3.3

Subtract $\sqrt{3 - x}$ from $- \sqrt{3 - x}$ .

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

Step 4

Solve for $- 2 \sqrt{3 - x}$ .

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Step 4.1

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

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

Step 4.2

Move all terms not containing $- 2 \sqrt{3 - x}$ to the right side of the equation.

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Step 4.2.1

Subtract $4$ from both sides of the equation.

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

Step 4.2.2

Add $x$ to both sides of the equation.

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

Step 4.2.3

Add $x$ and $x$ .

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

Step 4.2.4

Subtract $4$ from $2$ .

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

Step 5

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

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

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$4 {(3 - x)}^{\frac{1}{2} \cdot 2} = {(2 x - 2)}^{2}$

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.

$4 {(3 - x)}^{\frac{1}{2} \cdot 2} = {(2 x - 2)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$4 {(3 - x)}^{1} = {(2 x - 2)}^{2}$

Step 6.2.1.4

Simplify.

$4 (3 - x) = {(2 x - 2)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$4 \cdot 3 + 4 (- x) = {(2 x - 2)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $4$ by $3$ .

$12 + 4 (- x) = {(2 x - 2)}^{2}$

Step 6.2.1.6.2

Multiply $- 1$ by $4$ .

$12 - 4 x = {(2 x - 2)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

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

$12 - 4 x = (2 x - 2) (2 x - 2)$

Step 6.3.1.2

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

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Step 6.3.1.2.1

Apply the distributive property.

$12 - 4 x = 2 x (2 x - 2) - 2 (2 x - 2)$

Step 6.3.1.2.2

Apply the distributive property.

$12 - 4 x = 2 x (2 x) + 2 x \cdot - 2 - 2 (2 x - 2)$

Step 6.3.1.2.3

Apply the distributive property.

$12 - 4 x = 2 x (2 x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

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.

$12 - 4 x = 2 \cdot 2 x \cdot x + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

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 $x$ .

$12 - 4 x = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 6.3.1.3.1.3

Multiply $2$ by $2$ .

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

Step 6.3.1.3.1.4

Multiply $- 2$ by $2$ .

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

Step 6.3.1.3.1.5

Multiply $2$ by $- 2$ .

$12 - 4 x = 4 x^{2} - 4 x - 4 x - 2 \cdot - 2$

Step 6.3.1.3.1.6

Multiply $- 2$ by $- 2$ .

$12 - 4 x = 4 x^{2} - 4 x - 4 x + 4$

Step 6.3.1.3.2

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

$12 - 4 x = 4 x^{2} - 8 x + 4$

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.

$4 x^{2} - 8 x + 4 = 12 - 4 x$

Step 7.2

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

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Step 7.2.1

Add $4 x$ to both sides of the equation.

$4 x^{2} - 8 x + 4 + 4 x = 12$

Step 7.2.2

Add $- 8 x$ and $4 x$ .

$4 x^{2} - 4 x + 4 = 12$

Step 7.3

Subtract $12$ from both sides of the equation.

$4 x^{2} - 4 x + 4 - 12 = 0$

Step 7.4

Subtract $12$ from $4$ .

$4 x^{2} - 4 x - 8 = 0$

Step 7.5

Factor the left side of the equation.

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Step 7.5.1

Factor $4$ out of $4 x^{2} - 4 x - 8$ .

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Step 7.5.1.1

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

$4 (x^{2}) - 4 x - 8 = 0$

Step 7.5.1.2

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

$4 (x^{2}) + 4 (- x) - 8 = 0$

Step 7.5.1.3

Factor $4$ out of $- 8$ .

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

Step 7.5.1.4

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

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

Step 7.5.1.5

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

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

Step 7.5.2

Factor.

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Step 7.5.2.1

Factor $x^{2} - x - 2$ using the AC method.

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Step 7.5.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 7.5.2.1.2

Write the factored form using these integers.

$4 ((x - 2) (x + 1)) = 0$

Step 7.5.2.2

Remove unnecessary parentheses.

$4 (x - 2) (x + 1) = 0$

Step 7.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 1 = 0$

Step 7.7

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 7.7.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 7.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 7.8

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 7.8.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 7.8.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7.9

The final solution is all the values that make $4 (x - 2) (x + 1) = 0$ true.

$x = 2, - 1$

Step 8

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

$x = - 1$