If $z$ satisfies $z^{2} + 4 z + 5 = 0$ , compute $∣ z ∣$ .

medium 2 marks Class 11 · Mathematics · Complex Numbers and Quadratic Equations

Answer

5

The correct answer is $5$ — Both roots have the same modulus $s q r t 5$ .

Step-by-step solution

Write the quadratic in standard form $a z^{2} + b z + c = 0$ with $a = 1,; b = 4,; c = 5$ .
Use the quadratic formula: $z = \frac{- b p m s q r t b ^{2} - 4 a c}{2 a} = \frac{- 4 p m s q r t 16 - 20}{2} = \frac{- 4 p m 2 i}{2} = - 2 p mi$ .
Compute the modulus of $- 2 + i$ : $∣ z ∣ = (- 2)^{2} + 1^{2} = 4 + 1 = s q r t 5$ .
Both roots have the same modulus $s q r t 5$ .

Common mistake: Using only the real part of the root to compute the modulus, ignoring the imaginary part

Concept tested: Modulus of a complex root of a quadratic