Basic Math Examples

${(2 - \sqrt{y + 2})}^{2} = 4$

Step 1

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

$2 - \sqrt{y + 2} = \pm \sqrt{4}$

Step 2

Simplify $\pm \sqrt{4}$ .

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Step 2.1

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

$2 - \sqrt{y + 2} = \pm \sqrt{2^{2}}$

Step 2.2

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

$2 - \sqrt{y + 2} = \pm 2$

Step 3

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

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Step 3.1

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

$2 - \sqrt{y + 2} = 2$

Step 3.2

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

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Step 3.2.1

Subtract $2$ from both sides of the equation.

$- \sqrt{y + 2} = 2 - 2$

Step 3.2.2

Subtract $2$ from $2$ .

$- \sqrt{y + 2} = 0$

Step 3.3

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

${(- \sqrt{y + 2})}^{2} = 0^{2}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

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

${(- {(y + 2)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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

${(- 1)}^{2} {({(y + 2)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(y + 2)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.1.3

Multiply ${({(y + 2)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(y + 2)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.1.4

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

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Step 3.4.2.1.4.1

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

${(y + 2)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.4.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.4.2.1

Cancel the common factor.

${(y + 2)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.4.2.1.4.2.2

Rewrite the expression.

${(y + 2)}^{1} = 0^{2}$

Step 3.4.2.1.5

Simplify.

$y + 2 = 0^{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Raising $0$ to any positive power yields $0$ .

$y + 2 = 0$

Step 3.5

Subtract $2$ from both sides of the equation.

$y = - 2$

Step 3.6

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

$2 - \sqrt{y + 2} = - 2$

Step 3.7

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

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Step 3.7.1

Subtract $2$ from both sides of the equation.

$- \sqrt{y + 2} = - 2 - 2$

Step 3.7.2

Subtract $2$ from $- 2$ .

$- \sqrt{y + 2} = - 4$

Step 3.8

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

${(- \sqrt{y + 2})}^{2} = {(- 4)}^{2}$

Step 3.9

Simplify each side of the equation.

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Step 3.9.1

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

${(- {(y + 2)}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

Step 3.9.2

Simplify the left side.

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Step 3.9.2.1

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

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Step 3.9.2.1.1

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

${(- 1)}^{2} {({(y + 2)}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

Step 3.9.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(y + 2)}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

Step 3.9.2.1.3

Multiply ${({(y + 2)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(y + 2)}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

Step 3.9.2.1.4

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

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Step 3.9.2.1.4.1

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

${(y + 2)}^{\frac{1}{2} \cdot 2} = {(- 4)}^{2}$

Step 3.9.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.9.2.1.4.2.1

Cancel the common factor.

${(y + 2)}^{\frac{1}{2} \cdot 2} = {(- 4)}^{2}$

Step 3.9.2.1.4.2.2

Rewrite the expression.

${(y + 2)}^{1} = {(- 4)}^{2}$

Step 3.9.2.1.5

Simplify.

$y + 2 = {(- 4)}^{2}$

Step 3.9.3

Simplify the right side.

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Step 3.9.3.1

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

$y + 2 = 16$

Step 3.10

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

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Step 3.10.1

Subtract $2$ from both sides of the equation.

$y = 16 - 2$

Step 3.10.2

Subtract $2$ from $16$ .

$y = 14$

Step 3.11

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

$y = - 2, 14$