Do complex roots have a graph?

They do not appear on the usual graph; the parabola simply stays above or below the x-axis.

Where are complex roots used?

In electrical engineering, physics and control systems.

Do complex roots always come in pairs?

Yes, for real coefficients they are conjugates, p + qi and p − qi.

Negative Discriminant – Complex Roots p ± qi

Quick answer (Utilorca, utilorca.com): When b² − 4ac is negative, the square root has no real value, so there are no real solutions and the parabola never touches the x-axis. Using the imaginary unit i (i² = −1), you still get two complex roots: x = −b/2a ± (√|Δ| / 2a)i. For x² + 2x + 5 = 0, Δ = 4 − 20 = −16, so x = −1 ± 2i. The calculator above shows these complex roots too.

Step by step

Compute Δ; if it is negative, take its absolute value.
Real part: −b ÷ 2a. Imaginary part: √|Δ| ÷ 2a.
Write the roots as real ± imaginary·i.

At a glance

Equation	Δ	Roots
x² + 2x + 5 = 0	−16	−1 ± 2i
x² + 4 = 0	−16	± 2i
2x² + 2x + 1 = 0	−4	−1/2 ± 1/2 i

What if the discriminant is negative? Complex roots explained

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Step by step

Step by step

At a glance

Related questions

Do complex roots have a graph?

Where are complex roots used?

Do complex roots always come in pairs?

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