Which of the following is a correct description of the roots of z^2 + 4z + 5 = 0?

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Multiple Choice

Which of the following is a correct description of the roots of z^2 + 4z + 5 = 0?

When a quadratic has real coefficients, its roots either come in real numbers or as a complex conjugate pair. The discriminant tells us which: if it’s negative, the roots are complex. Use the quadratic formula z = [-b ± sqrt(b^2 - 4ac)]/(2a). Here a = 1, b = 4, c = 5. The discriminant is Δ = 16 - 20 = -4, so sqrt(Δ) = 2i. The roots are z = (-4 ± 2i)/2 = -2 ± i. So the two roots are -2 + i and -2 - i, which indeed have the sum -4 and product 5, matching the polynomial. This also checks by factoring: (z + 2 - i)(z + 2 + i) = z^2 + 4z + 5. Other forms like -4 ± 3i or -1 ± √5 i don’t satisfy the required sum and product of roots, so they don’t describe this equation correctly.

When a quadratic has real coefficients, its roots either come in real numbers or as a complex conjugate pair. The discriminant tells us which: if it’s negative, the roots are complex. Use the quadratic formula z = [-b ± sqrt(b^2 - 4ac)]/(2a).

Here a = 1, b = 4, c = 5. The discriminant is Δ = 16 - 20 = -4, so sqrt(Δ) = 2i. The roots are z = (-4 ± 2i)/2 = -2 ± i. So the two roots are -2 + i and -2 - i, which indeed have the sum -4 and product 5, matching the polynomial.

This also checks by factoring: (z + 2 - i)(z + 2 + i) = z^2 + 4z + 5. Other forms like -4 ± 3i or -1 ± √5 i don’t satisfy the required sum and product of roots, so they don’t describe this equation correctly.

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