Solve {c}{f(z)=(10+i)z^2}{z=7+i}

$$\left. \begin{array} { c } { f ( z ) = ( 10 + i ) z ^ { 2 } } \\ { z = 7 + i } \end{array} \right.$$

Solve for f, z

$$z=7+i$$

$$f=69+17i$$

Solution Steps

Consider the first equation. Insert the known values of variables into the equation.

$$f\left(7+i\right)=\left(10+i\right)\left(7+i\right)^{2}$$

Calculate 7+i to the power of 2 and get 48+14i.

$$f\left(7+i\right)=\left(10+i\right)\left(48+14i\right)$$

Multiply 10+i and 48+14i to get 466+188i.

$$f\left(7+i\right)=466+188i$$

Divide both sides by 7+i.

$$f=\frac{466+188i}{7+i}$$

Multiply both numerator and denominator of \frac{466+188i}{7+i} by the complex conjugate of the denominator, 7-i.

$$f=\frac{\left(466+188i\right)\left(7-i\right)}{\left(7+i\right)\left(7-i\right)}$$

Do the multiplications in \frac{\left(466+188i\right)\left(7-i\right)}{\left(7+i\right)\left(7-i\right)}.

$$f=\frac{3450+850i}{50}$$

Divide 3450+850i by 50 to get 69+17i.

$$f=69+17i$$

The system is now solved.

$$f=69+17i z=7+i$$