Given z1 = 2+3i and z2 = −3+i, find the real part of the product z1 z2.

Question

ScanSolve AI Tutor · Accepted Answer

Step 1: Identify the given complex numbers. We are given two complex numbers: z₁ = 2+3i; z₂ = −3+i Step 2: Calculate the product z₁ z₂. Multiply the two complex numbers: z₁ z₂ = (2+3i)(−3+i) Use the distributive property (FOIL method): z₁ z₂ = 2(−3) + 2(i) + 3i(−3) + 3i(i); z₁ z₂ = −6 + 2i − 9i + 3i²; Step 3: Substitute i² = −1. Recall that i² = −1. Substitute this into the expression: z₁ z₂ = −6 + 2i − 9i + 3(−1); z₁ z₂ = −6 + 2i − 9i − 3 Step 4: Combine the real and imaginary parts. Group the real terms and the imaginary terms: z₁ z₂ = (−6 − 3) + (2i − 9i); z₁ z₂ = −9 − 7i Step 5: Identify the real part of z₁ z₂. The real part of a complex number a+bi is a. In this case, the real part of −9−7i is −9. The final answer is −9.

Given z1 = 2+3i and z2 = -3+i, find the real part of the product z1 z2.

Answer

Step-by-step solution

Identify the given complex numbers.

Calculate the product $z_1 z_2$ .

Substitute $i^2 = -1$ .

Combine the real and imaginary parts.

Identify the real part of $z_1 z_2$ .

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