Algebra Examples

$r = \frac{1}{2 A} \sqrt{B^{2} + C^{2} - 4 A F}$

Step 1

Rewrite the equation as $\frac{1}{2 A} \cdot \sqrt{B^{2} + C^{2} - 4 A F} = r$ .

$\frac{1}{2 A} \cdot \sqrt{B^{2} + C^{2} - 4 A F} = r$

Step 2

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

${(\frac{1}{2 A} \cdot \sqrt{B^{2} + C^{2} - 4 A F})}^{2} = r^{2}$

Step 3

Simplify each side of the equation.

Tap for more steps...

Step 3.1

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

${(\frac{1}{2 A} \cdot {(B^{2} + C^{2} - 4 A F)}^{\frac{1}{2}})}^{2} = r^{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${(\frac{1}{2 A} \cdot {(B^{2} + C^{2} - 4 A F)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1

Combine $\frac{1}{2 A}$ and ${(B^{2} + C^{2} - 4 A F)}^{\frac{1}{2}}$ .

${(\frac{{(B^{2} + C^{2} - 4 A F)}^{\frac{1}{2}}}{2 A})}^{2} = r^{2}$

Step 3.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 3.2.1.2.1

Apply the product rule to $\frac{{(B^{2} + C^{2} - 4 A F)}^{\frac{1}{2}}}{2 A}$ .

$\frac{{({(B^{2} + C^{2} - 4 A F)}^{\frac{1}{2}})}^{2}}{{(2 A)}^{2}} = r^{2}$

Step 3.2.1.2.2

Apply the product rule to $2 A$ .

$\frac{{({(B^{2} + C^{2} - 4 A F)}^{\frac{1}{2}})}^{2}}{2^{2} A^{2}} = r^{2}$

Step 3.2.1.3

Simplify the numerator.

Tap for more steps...

Step 3.2.1.3.1

Multiply the exponents in ${({(B^{2} + C^{2} - 4 A F)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.3.1.1

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

$\frac{{(B^{2} + C^{2} - 4 A F)}^{\frac{1}{2} \cdot 2}}{2^{2} A^{2}} = r^{2}$

Step 3.2.1.3.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.3.1.2.1

Cancel the common factor.

$\frac{{(B^{2} + C^{2} - 4 A F)}^{\frac{1}{2} \cdot 2}}{2^{2} A^{2}} = r^{2}$

Step 3.2.1.3.1.2.2

Rewrite the expression.

$\frac{{(B^{2} + C^{2} - 4 A F)}^{1}}{2^{2} A^{2}} = r^{2}$

Step 3.2.1.3.2

Simplify.

$\frac{B^{2} + C^{2} - 4 A F}{2^{2} A^{2}} = r^{2}$

Step 3.2.1.4

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

$\frac{B^{2} + C^{2} - 4 A F}{4 A^{2}} = r^{2}$

Step 4

Solve for $F$ .

Tap for more steps...

Step 4.1

Multiply both sides by $4 A^{2}$ .

$\frac{B^{2} + C^{2} - 4 A F}{4 A^{2}} (4 A^{2}) = r^{2} (4 A^{2})$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

Simplify the left side.

Tap for more steps...

Step 4.2.1.1

Simplify $\frac{B^{2} + C^{2} - 4 A F}{4 A^{2}} (4 A^{2})$ .

Tap for more steps...

Step 4.2.1.1.1

Rewrite using the commutative property of multiplication.

$4 \frac{B^{2} + C^{2} - 4 A F}{4 A^{2}} A^{2} = r^{2} (4 A^{2})$

Step 4.2.1.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.2.1.1.2.1

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

$4 \frac{B^{2} + C^{2} - 4 A F}{4 (A^{2})} A^{2} = r^{2} (4 A^{2})$

Step 4.2.1.1.2.2

Cancel the common factor.

$4 \frac{B^{2} + C^{2} - 4 A F}{4 A^{2}} A^{2} = r^{2} (4 A^{2})$

Step 4.2.1.1.2.3

Rewrite the expression.

$\frac{B^{2} + C^{2} - 4 A F}{A^{2}} A^{2} = r^{2} (4 A^{2})$

Step 4.2.1.1.3

Cancel the common factor of $A^{2}$ .

Tap for more steps...

Step 4.2.1.1.3.1

Cancel the common factor.

$\frac{B^{2} + C^{2} - 4 A F}{A^{2}} A^{2} = r^{2} (4 A^{2})$

Step 4.2.1.1.3.2

Rewrite the expression.

$B^{2} + C^{2} - 4 A F = r^{2} (4 A^{2})$

Step 4.2.1.1.4

Simplify the expression.

Tap for more steps...

Step 4.2.1.1.4.1

Move $C^{2}$ .

$B^{2} - 4 A F + C^{2} = r^{2} (4 A^{2})$

Step 4.2.1.1.4.2

Reorder $B^{2}$ and $- 4 A F$ .

$- 4 A F + B^{2} + C^{2} = r^{2} (4 A^{2})$

Step 4.2.2

Simplify the right side.

Tap for more steps...

Step 4.2.2.1

Rewrite using the commutative property of multiplication.

$- 4 A F + B^{2} + C^{2} = 4 r^{2} A^{2}$

Step 4.3

Solve for $F$ .

Tap for more steps...

Step 4.3.1

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

Tap for more steps...

Step 4.3.1.1

Subtract $B^{2}$ from both sides of the equation.

$- 4 A F + C^{2} = 4 r^{2} A^{2} - B^{2}$

Step 4.3.1.2

Subtract $C^{2}$ from both sides of the equation.

$- 4 A F = 4 r^{2} A^{2} - B^{2} - C^{2}$

Step 4.3.2

Divide each term in $- 4 A F = 4 r^{2} A^{2} - B^{2} - C^{2}$ by $- 4 A$ and simplify.

Tap for more steps...

Step 4.3.2.1

Divide each term in $- 4 A F = 4 r^{2} A^{2} - B^{2} - C^{2}$ by $- 4 A$ .

$\frac{- 4 A F}{- 4 A} = \frac{4 r^{2} A^{2}}{- 4 A} + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{- 4 A F}{- 4 A} = \frac{4 r^{2} A^{2}}{- 4 A} + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.2.1.2

Rewrite the expression.

$\frac{A F}{A} = \frac{4 r^{2} A^{2}}{- 4 A} + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.2.2

Cancel the common factor of $A$ .

Tap for more steps...

Step 4.3.2.2.2.1

Cancel the common factor.

$\frac{A F}{A} = \frac{4 r^{2} A^{2}}{- 4 A} + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.2.2.2

Divide $F$ by $1$ .

$F = \frac{4 r^{2} A^{2}}{- 4 A} + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.3

Simplify the right side.

Tap for more steps...

Step 4.3.2.3.1

Simplify each term.

Tap for more steps...

Step 4.3.2.3.1.1

Cancel the common factor of $4$ and $- 4$ .

Tap for more steps...

Step 4.3.2.3.1.1.1

Factor $4$ out of $4 r^{2} A^{2}$ .

$F = \frac{4 (r^{2} A^{2})}{- 4 A} + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 4.3.2.3.1.1.2.1

Factor $4$ out of $- 4 A$ .

$F = \frac{4 (r^{2} A^{2})}{4 (- A)} + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.3.1.1.2.2

Cancel the common factor.

$F = \frac{4 (r^{2} A^{2})}{4 (- A)} + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.3.1.1.2.3

Rewrite the expression.

$F = \frac{r^{2} A^{2}}{- A} + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.3.1.2

Cancel the common factor of $A^{2}$ and $A$ .

Tap for more steps...

Step 4.3.2.3.1.2.1

Factor $A$ out of $r^{2} A^{2}$ .

$F = \frac{A (r^{2} A)}{- A} + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.3.1.2.2

Move the negative one from the denominator of $\frac{r^{2} A}{- 1}$ .

$F = - 1 \cdot (r^{2} A) + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.3.1.3

Rewrite $- 1 \cdot (r^{2} A)$ as $- (r^{2} A)$ .

$F = - (r^{2} A) + \frac{- B^{2}}{- 4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.3.1.4

Dividing two negative values results in a positive value.

$F = - r^{2} A + \frac{B^{2}}{4 A} + \frac{- C^{2}}{- 4 A}$

Step 4.3.2.3.1.5

Dividing two negative values results in a positive value.

$F = - r^{2} A + \frac{B^{2}}{4 A} + \frac{C^{2}}{4 A}$