Evaluate |48+bi|=50

$| 48 + b i | = 50$

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$48 + b i = \pm 50$

Step 2

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

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

$48 + b i = 50$

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

Subtract $48$ from both sides of the equation.

$b i = 50 - 48$

Subtract $48$ from $50$ .

$b i = 2$

Divide each term in $b i = 2$ by $i$ and simplify.

Divide each term in $b i = 2$ by $i$ .

$\frac{b i}{i} = \frac{2}{i}$

Simplify the left side.

Cancel the common factor of $i$ .

Cancel the common factor.

$\frac{b i}{i} = \frac{2}{i}$

Divide $b$ by $1$ .

$b = \frac{2}{i}$

Simplify the right side.

Multiply the numerator and denominator of $\frac{2}{i}$ by the conjugate of $i$ to make the denominator real.

$b = \frac{2}{i} \cdot \frac{i}{i}$

Multiply.

Combine.

$b = \frac{2 i}{i i}$

Simplify the denominator.

Raise $i$ to the power of $1$ .

$b = \frac{2 i}{i^{1} i}$

Raise $i$ to the power of $1$ .

$b = \frac{2 i}{i^{1} i^{1}}$

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

$b = \frac{2 i}{i^{1 + 1}}$

Add $1$ and $1$ .

$b = \frac{2 i}{i^{2}}$

Rewrite $i^{2}$ as $- 1$ .

$b = \frac{2 i}{- 1}$

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

$b = - 1 \cdot (2 i)$

Rewrite $- 1 \cdot (2 i)$ as $- (2 i)$ .

$b = - (2 i)$

Multiply $2$ by $- 1$ .

$b = - 2 i$

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

$48 + b i = - 50$

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

Subtract $48$ from both sides of the equation.

$b i = - 50 - 48$

Subtract $48$ from $- 50$ .

$b i = - 98$

Divide each term in $b i = - 98$ by $i$ and simplify.

Divide each term in $b i = - 98$ by $i$ .

$\frac{b i}{i} = \frac{- 98}{i}$

Simplify the left side.

Cancel the common factor of $i$ .

Cancel the common factor.

$\frac{b i}{i} = \frac{- 98}{i}$

Divide $b$ by $1$ .

$b = \frac{- 98}{i}$

Simplify the right side.

Multiply the numerator and denominator of $\frac{- 98}{i}$ by the conjugate of $i$ to make the denominator real.

$b = \frac{- 98}{i} \cdot \frac{i}{i}$

Multiply.

Combine.

$b = \frac{- 98 i}{i i}$

Simplify the denominator.

Raise $i$ to the power of $1$ .

$b = \frac{- 98 i}{i^{1} i}$

Raise $i$ to the power of $1$ .

$b = \frac{- 98 i}{i^{1} i^{1}}$

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

$b = \frac{- 98 i}{i^{1 + 1}}$

Add $1$ and $1$ .

$b = \frac{- 98 i}{i^{2}}$

Rewrite $i^{2}$ as $- 1$ .

$b = \frac{- 98 i}{- 1}$

Move the negative one from the denominator of $\frac{- 98 i}{- 1}$ .

$b = - 1 \cdot (- 98 i)$

Rewrite $- 1 \cdot (- 98 i)$ as $- (- 98 i)$ .

$b = - (- 98 i)$

Multiply $- 98$ by $- 1$ .

$b = 98 i$

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

$b = - 2 i, 98 i$

Evaluate |48+bi|=50

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