Non-real roots of quadratic equations

Japanese school year: Math II

What you learn

This section explores finding complex roots for quadratic equations when the discriminant is negative by extending the quadratic formula. It ensures that every quadratic equation has solutions, leading to deeper algebraic concepts. Before starting, you should be familiar with the real quadratic formula and basic complex arithmetic.

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Key points

This rule defines the square root of a negative number. When $a > 0$ , the negative sign inside the square root is extracted as the imaginary unit $i$ , written as $\sqrt{a}\,i$ .

\sqrt{-a} = \sqrt{a}\,i \quad (a > 0)

This condition determines whether a quadratic equation $ax^2+bx+c=0$ has imaginary solutions instead of real roots. When the discriminant $D=b^2-4ac$ is negative ( $D < 0$ ), the equation yields two distinct complex solutions involving $i$ .

D = b^2 - 4ac < 0

This quadratic formula finds the solutions when the discriminant is negative ( $b^2-4ac < 0$ ). It flips the sign inside the square root to positive $4ac-b^2$ and brings the imaginary unit $i$ outside.

x = \frac{-b \pm i\sqrt{4ac-b^2}}{2a} \quad (b^2-4ac < 0)

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Practice 1Coming soon