Precalculus Examples

$- 4 - \sqrt{4 - x} = 6$

Step 1

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

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Step 1.1

Add $4$ to both sides of the equation.

$- \sqrt{4 - x} = 6 + 4$

Step 1.2

Add $6$ and $4$ .

$- \sqrt{4 - x} = 10$

Step 2

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

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

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 $\sqrt{4 - x}$ as ${(4 - x)}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Multiply ${({(4 - x)}^{\frac{1}{2}})}^{2}$ by $1$ .

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

Step 3.2.1.4

Multiply the exponents in ${({(4 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.4.1

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

${(4 - x)}^{\frac{1}{2} \cdot 2} = 10^{2}$

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

${(4 - x)}^{\frac{1}{2} \cdot 2} = 10^{2}$

Step 3.2.1.4.2.2

Rewrite the expression.

${(4 - x)}^{1} = 10^{2}$

Step 3.2.1.5

Simplify.

$4 - x = 10^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $10$ to the power of $2$ .

$4 - x = 100$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $4$ from both sides of the equation.

$- x = 100 - 4$

Step 4.1.2

Subtract $4$ from $100$ .

$- x = 96$

Step 4.2

Divide each term in $- x = 96$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- x = 96$ by $- 1$ .

$\frac{- x}{- 1} = \frac{96}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{96}{- 1}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{96}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $96$ by $- 1$ .

$x = - 96$

Step 5

Exclude the solutions that do not make $- 4 - \sqrt{4 - x} = 6$ true.

$No solution$