Problem 75 - College Algebra Essentials

Question

Perform the indicated operations and write the result in standard form. $$ \frac{4}{(2+i)(3-i)} $$

Step-by-Step Solution

Answer

The simplified form of the given complex fraction is \(1.75 - 0.5i\).

Step 1: Identify the Conjugate of the Denominator

The conjugate of the denominator, which is \((2 + i)(3 - i)\), is \((2 - i)(3 + i)\). The conjugate of any complex number \((a + bi)\) is \((a - bi)\).

Step 2: Multiply the Numerator and Denominator by the Conjugate of the Denominator

To simplify the complex fraction, multiply both the numerator and the denominator by the conjugate of the denominator, \((2 - i)(3 + i)\). This gives \(\frac{4(2 - i)(3 + i)}{(2 + i)(3 - i)(2 - i)(3 + i)}\).

Step 3: Simplify the Numerator and the Denominator Separately

Expand the expression in the numerator and the denominators separately.\nThe numerator expands to \(4(6 + i - 3i - i^2)\) = \(4(6 - 2i + 1)\) = \(4(7 - 2i)\), which simplifies to \(28 - 8i\). \nThe denominator expands to \(6 - 2i + 3i + i^2 - 2i + 4 + 2i^2 - 3i + 2i + 4 - i^2\) = \(14 + 0i - 2i^2 = 14 + 2 = 16\).

Step 4: Write the Final Answer in Standard Form

Finally, divide the numerator by the denominator to find the quotient: \(\frac{28 - 8i}{16}\) = \(1.75 - 0.5i\). This is the final answer in the standard form.