Solve the complex equation.

01 / Standard#Your turn

z-3\bar z+|z|=-1+16i

Official paper · jm02-2023 · 4(a) · PDF 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Set z=x+yi and compare imaginary parts first.

Hint 2

Before squaring the real equation, require its right side nonnegative.

Worked solution

Separate the equations.
$4y=16\implies y=4,\quad\sqrt{x^2+16}=2x-1\ge0\implies x\ge1/2$
Square and factor.
$x^2+16=(2x-1)^2\iff(x-3)(3x+5)=0$
Only x=3 satisfies the unsquared equation.
$z=3+4i,\quad|z|=5$

z=3+4i.

Checks and common pitfalls: The root x=−5/3 introduced by squaring is extraneous.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Separate the equations.
$4y=16\implies y=4,\quad\sqrt{x^2+16}=2x-1\ge0\implies x\ge1/2$
Square and factor.
$x^2+16=(2x-1)^2\iff(x-3)(3x+5)=0$
Only x=3 satisfies the unsquared equation.
$z=3+4i,\quad|z|=5$

Think first. Reveal a hint when the class is ready.

2023 JM02