Perform each operation. Write answers in standard form. (-12 -i) / (-2 -5i)

Verified step by step guidance

1. Identify the given expression: \(\frac{-12 - i}{-2 - 5i}\). Our goal is to simplify this complex fraction and write the answer in standard form \(a + bi\), where \(a\) and \(b\) are real numbers.
2. To simplify a complex fraction, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(-2 - 5i\) is \(-2 + 5i\). So, multiply numerator and denominator by \(-2 + 5i\):
3. \[\frac{-12 - i}{-2 - 5i} \times \frac{-2 + 5i}{-2 + 5i}\]
4. Expand the numerator using the distributive property (FOIL): multiply \((-12)\) by \((-2 + 5i)\) and \((-i)\) by \((-2 + 5i)\), then combine like terms.
5. Expand the denominator using the difference of squares formula: \((a - bi)(a + bi) = a^2 + b^2\). Here, calculate \((-2)^2 + (5)^2\) to get a real number denominator.

Key Concepts

Complex Number Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. Writing answers in standard form means presenting the result clearly as a sum of a real number and an imaginary number.

Division of Complex Numbers

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. This process eliminates the imaginary part in the denominator, allowing the expression to be simplified into standard form.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying by the conjugate helps remove imaginary terms from denominators, making it easier to simplify complex fractions into standard form.