I May Be Irrational, But You’re Imaginary

Learning Focus

Find non-real solutions of quadratics using $i=\sqrt{-1}$.

Big idea

Define $i^2=-1$. For $c>0$: $\sqrt{-c}=i\sqrt c$.

Real irrational numbers (like $\sqrt2$) are still real; imaginary numbers (like $3i$) are not. When $D<0$ the formula gives:

$x=\frac{-b}{2a}\pm\frac{\sqrt{|D|}}{2a}\,i.$

Worked example

$x^2-2x+5=0$: $D=4-20=-16$.

$x=\frac{2\pm\sqrt{-16}}{2}=\frac{2\pm4i}{2}=1\pm2i.$

Takeaways

• Irrational and imaginary are different.
• Negative discriminant gives a conjugate pair $p\pm qi$.
• Complex solutions are not $x$-intercepts.