If i represents the imaginary unit, and the complex number z satisfies the equation z(2 + … — ACT

If $i$ represents the imaginary unit, and the complex number $z$ satisfies the equation $z(2 + i) - 5i^{97} = \frac{25}{3 - 4i}$ , what is the value of $z^2$ ?

A
-18i
B
-2i
C
16
18iAnswer
E
50i

Answer

18i

Simplifying the original equation yields the complex number

z = 3 + 3i

. Squaring this number gives

(3 + 3i)^2 = 9 + 18i + 9i^2 = 9 + 18i - 9 = 18i

Step-by-Step Solution

Simplify the fraction on the right side of the equation by multiplying the numerator and denominator by the complex conjugate of the denominator,

3 + 4i

The fraction simplifies to

3 + 4i

Multiplying by the conjugate rationalizes the denominator:

\frac{25}{3 - 4i} = \frac{25(3 + 4i)}{(3 - 4i)(3 + 4i)} = \frac{25(3 + 4i)}{9 + 16} = 3 + 4i

Simplify the power of the imaginary unit,

i^{97}

, by dividing the exponent by 4 to find the remainder.

5i^{97} = 5i

Since

97 = 4 \times 24 + 1

, the expression simplifies as

i^{97} = (i^4)^{24} \cdot i = 1^{24} \cdot i = i

Substitute the simplified expressions back into the original equation and isolate the term containing

z

z(2 + i) = 3 + 9i

Substituting gives

z(2 + i) - 5i = 3 + 4i

. Adding

5i

to both sides yields

z(2 + i) = 3 + 9i

Solve for

z

by dividing both sides by

2 + i

, then simplify by multiplying by the conjugate of the denominator,

2 - i

z = 3 + 3i

Performing the division:

z = \frac{3 + 9i}{2 + i} = \frac{(3 + 9i)(2 - i)}{(2 + i)(2 - i)} = \frac{6 - 3i + 18i - 9i^2}{4 - i^2} = \frac{15 + 15i}{5} = 3 + 3i

Calculate the value of

z^2

by squaring the complex number

3 + 3i

z^2 = 18i

Squaring the binomial gives

(3 + 3i)^2 = 9 + 18i + 9i^2 = 9 + 18i - 9 = 18i

Key Concept

Solving equations containing complex numbers by performing operations such as multiplication, division using complex conjugates, and simplifying powers of

i

Estimated Time:2m 30s

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