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Worked solution

Find paired real unknowns by comparing components

Given that (2−i)2+(3λ+i)(μ−i)+5i=10(2-i)^2+(3\lambda+i)(\mu-i)+5i=10, find the exact values of the real numbers λ\lambda and μ\mu.

CJC · Example 3

Answer

(λ,μ)=(1,2)(\lambda,\mu)=(1,2) or (−23,−3)\left(-\frac23,-3\right).

Compare your reasoning

Expand the products, use i2=−1i^2=-1, then collect real and imaginary terms:

[2^2-2(2)i+i^2]+[3\lambda\mu-3\lambda i+\mu i-i^2]+5i&=10,\\ 4-4i-1+3\lambda\mu-3\lambda i+\mu i+1+5i&=10,\\ 4+3\lambda\mu+(\mu-3\lambda+1)i&=10. \end{aligned}$$ The key method is to compare real parts and imaginary coefficients separately. The right side is $10+0i$, so $$\begin{aligned}4+3\lambda\mu&=10 &&(1),\\ \mu-3\lambda+1&=0 &&(2). \end{aligned}$$ From (2), $\mu=3\lambda-1$ (3). Substitute (3) into (1): $$\begin{aligned} 4+3\lambda(3\lambda-1)&=10,\\ 9\lambda^2-3\lambda-6&=0,\\ 3\lambda^2-\lambda-2&=0,\\ (3\lambda+2)(\lambda-1)&=0. \end{aligned}$$ Hence $\lambda=-\frac23$ or $\lambda=1$. Use (3) for each value: when $\lambda=1$, $\mu=2$; when $\lambda=-\frac23$, $\mu=-3$. Keep each value paired with its corresponding partner.
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Solution notes

The Teacher emphasis on comparing real and imaginary parts is integrated into the method above. Both copies print the full derivation. Do not combine a value of λ\lambda from one branch with a value of μ\mu from the other.