Solve 3-1/(2-i)(5+2i)

$$\frac { 3 - 1 } { ( 2 - i ) ( 5 + 2 i ) }$$

Evaluate

$$\frac{24}{145}+\frac{2}{145}i\approx 0.165517241+0.013793103i$$

Solution Steps

Subtract 1 from 3 to get 2.

$$\frac{2}{\left(2-i\right)\left(5+2i\right)}$$

Multiply complex numbers 2-i and 5+2i like you multiply binomials.

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

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

$$\frac{2}{2\times 5+2\times \left(2i\right)-i\times 5-2\left(-1\right)}$$

Do the multiplications in 2\times 5+2\times \left(2i\right)-i\times 5-2\left(-1\right).

$$\frac{2}{10+4i-5i+2}$$

Combine the real and imaginary parts in 10+4i-5i+2.

$$\frac{2}{10+2+\left(4-5\right)i}$$

Do the additions in 10+2+\left(4-5\right)i.

$$\frac{2}{12-i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 12+i.

$$\frac{2\left(12+i\right)}{\left(12-i\right)\left(12+i\right)}$$

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

$$\frac{2\left(12+i\right)}{12^{2}-i^{2}}$$

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

$$\frac{2\left(12+i\right)}{145}$$

Multiply 2 times 12+i.

$$\frac{2\times 12+2i}{145}$$

Do the multiplications in 2\times 12+2i.

$$\frac{24+2i}{145}$$

Divide 24+2i by 145 to get \frac{24}{145}+\frac{2}{145}i.

$$\frac{24}{145}+\frac{2}{145}i$$

Real Part

$$\frac{24}{145} = 0.16551724137931034$$

Solution Steps

Subtract 1 from 3 to get 2.

$$Re(\frac{2}{\left(2-i\right)\left(5+2i\right)})$$

Multiply complex numbers 2-i and 5+2i like you multiply binomials.

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

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

$$Re(\frac{2}{2\times 5+2\times \left(2i\right)-i\times 5-2\left(-1\right)})$$

Do the multiplications in 2\times 5+2\times \left(2i\right)-i\times 5-2\left(-1\right).

$$Re(\frac{2}{10+4i-5i+2})$$

Combine the real and imaginary parts in 10+4i-5i+2.

$$Re(\frac{2}{10+2+\left(4-5\right)i})$$

Do the additions in 10+2+\left(4-5\right)i.

$$Re(\frac{2}{12-i})$$

Multiply both numerator and denominator of \frac{2}{12-i} by the complex conjugate of the denominator, 12+i.

$$Re(\frac{2\left(12+i\right)}{\left(12-i\right)\left(12+i\right)})$$

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

$$Re(\frac{2\left(12+i\right)}{12^{2}-i^{2}})$$

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

$$Re(\frac{2\left(12+i\right)}{145})$$

Multiply 2 times 12+i.

$$Re(\frac{2\times 12+2i}{145})$$

Do the multiplications in 2\times 12+2i.

$$Re(\frac{24+2i}{145})$$

Divide 24+2i by 145 to get \frac{24}{145}+\frac{2}{145}i.

$$Re(\frac{24}{145}+\frac{2}{145}i)$$

The real part of \frac{24}{145}+\frac{2}{145}i is \frac{24}{145}.

$$\frac{24}{145}$$