Solve 24+11i/-4+5i

$$\frac { 24 + 11 i } { - 4 + 5 i }$$

Evaluate

$$-1-4i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -4-5i.

$$\frac{\left(24+11i\right)\left(-4-5i\right)}{\left(-4+5i\right)\left(-4-5i\right)}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(24+11i\right)\left(-4-5i\right)}{\left(-4\right)^{2}-5^{2}i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{\left(24+11i\right)\left(-4-5i\right)}{41}$$

Multiply complex numbers 24+11i and -4-5i like you multiply binomials.

$$\frac{24\left(-4\right)+24\times \left(-5i\right)+11i\left(-4\right)+11\left(-5\right)i^{2}}{41}$$

By definition, i^{2} is -1.

$$\frac{24\left(-4\right)+24\times \left(-5i\right)+11i\left(-4\right)+11\left(-5\right)\left(-1\right)}{41}$$

Do the multiplications in 24\left(-4\right)+24\times \left(-5i\right)+11i\left(-4\right)+11\left(-5\right)\left(-1\right).

$$\frac{-96-120i-44i+55}{41}$$

Combine the real and imaginary parts in -96-120i-44i+55.

$$\frac{-96+55+\left(-120-44\right)i}{41}$$

Do the additions in -96+55+\left(-120-44\right)i.

$$\frac{-41-164i}{41}$$

Divide -41-164i by 41 to get -1-4i.

$$-1-4i$$

Real Part

$$-1$$

Solution Steps

Multiply both numerator and denominator of \frac{24+11i}{-4+5i} by the complex conjugate of the denominator, -4-5i.

$$Re(\frac{\left(24+11i\right)\left(-4-5i\right)}{\left(-4+5i\right)\left(-4-5i\right)})$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(\frac{\left(24+11i\right)\left(-4-5i\right)}{\left(-4\right)^{2}-5^{2}i^{2}})$$

By definition, i^{2} is -1. Calculate the denominator.

$$Re(\frac{\left(24+11i\right)\left(-4-5i\right)}{41})$$

Multiply complex numbers 24+11i and -4-5i like you multiply binomials.

$$Re(\frac{24\left(-4\right)+24\times \left(-5i\right)+11i\left(-4\right)+11\left(-5\right)i^{2}}{41})$$

By definition, i^{2} is -1.

$$Re(\frac{24\left(-4\right)+24\times \left(-5i\right)+11i\left(-4\right)+11\left(-5\right)\left(-1\right)}{41})$$

Do the multiplications in 24\left(-4\right)+24\times \left(-5i\right)+11i\left(-4\right)+11\left(-5\right)\left(-1\right).

$$Re(\frac{-96-120i-44i+55}{41})$$

Combine the real and imaginary parts in -96-120i-44i+55.

$$Re(\frac{-96+55+\left(-120-44\right)i}{41})$$

Do the additions in -96+55+\left(-120-44\right)i.

$$Re(\frac{-41-164i}{41})$$

Divide -41-164i by 41 to get -1-4i.

$$Re(-1-4i)$$

The real part of -1-4i is -1.

$$-1$$