Solve for f f-2 = square root of f+4

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{f + 4}$ as ${(f + 4)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${({(f + 4)}^{\frac{1}{2}})}^{2}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$f + 4 = (f - 2) (f - 2)$

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

Apply the distributive property.

$f + 4 = f (f - 2) - 2 (f - 2)$

Apply the distributive property.

$f + 4 = f \cdot f + f \cdot - 2 - 2 (f - 2)$

Apply the distributive property.

$f + 4 = f \cdot f + f \cdot - 2 - 2 f - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $f$ by $f$ .

$f + 4 = f^{2} + f \cdot - 2 - 2 f - 2 \cdot - 2$

Move $- 2$ to the left of $f$ .

$f + 4 = f^{2} - 2 \cdot f - 2 f - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$f + 4 = f^{2} - 2 f - 2 f + 4$

Subtract $2 f$ from $- 2 f$ .

$f + 4 = f^{2} - 4 f + 4$

Step 4

Solve for $f$ .

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

$f^{2} - 4 f + 4 = f + 4$

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

Subtract $f$ from both sides of the equation.

$f^{2} - 4 f + 4 - f = 4$

Subtract $f$ from $- 4 f$ .

$f^{2} - 5 f + 4 = 4$

Subtract $4$ from both sides of the equation.

$f^{2} - 5 f + 4 - 4 = 0$

Combine the opposite terms in $f^{2} - 5 f + 4 - 4$ .

Subtract $4$ from $4$ .

$f^{2} - 5 f + 0 = 0$

Add $f^{2} - 5 f$ and $0$ .

$f^{2} - 5 f = 0$

Factor $f$ out of $f^{2} - 5 f$ .

Factor $f$ out of $f^{2}$ .

$f \cdot f - 5 f = 0$

Factor $f$ out of $- 5 f$ .

$f \cdot f + f \cdot - 5 = 0$

Factor $f$ out of $f \cdot f + f \cdot - 5$ .

$f (f - 5) = 0$

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

$f = 0$

$f - 5 = 0$

Set $f$ equal to $0$ .

$f = 0$

Set $f - 5$ equal to $0$ and solve for $f$ .

Set $f - 5$ equal to $0$ .

$f - 5 = 0$

Add $5$ to both sides of the equation.

$f = 5$

The final solution is all the values that make $f (f - 5) = 0$ true.

$f = 0, 5$

Step 5

Exclude the solutions that do not make $f - 2 = \sqrt{f + 4}$ true.

$f = 5$

Solve for f f-2 = square root of f+4

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