The discriminant and complex numbers
The expression under the square root in the quadratic formula predicts how many real solutions a quadratic has. When it is negative, the solutions are complex numbers.
The discriminant
The discriminant of \(ax^2 + bx + c\) is \(D = b^2 - 4ac\). If \(D > 0\), there are two different real solutions; if \(D = 0\), one repeated real solution; if \(D < 0\), no real solutions. For \(x^2 - 4x + 4\), \(D = 16 - 16 = 0\), and indeed \((x - 2)^2 = 0\) has only x = 2.
Discriminant and graphs
The discriminant matches the graph: D > 0 means the parabola crosses the x-axis twice, D = 0 means it touches the axis at its vertex, and D < 0 means it misses the axis entirely. A positive perfect-square discriminant, such as 25, also tells you the quadratic factors over the integers when a, b, and c are integers.
The imaginary unit
To solve equations like \(x^2 = -1\), mathematicians define the imaginary unit i with \(i^2 = -1\). Then \(\sqrt{-9} = 3i\) and \(\sqrt{-12} = 2i\sqrt{3}\). The term 'imaginary' was originally dismissive, but these numbers are essential in electrical engineering, signal processing, and quantum physics.
Complex numbers
A complex number has the form \(a + bi\), with real part a and imaginary part b. Add and subtract by combining parts: \((3 + 2i) + (1 - 5i) = 4 - 3i\). Multiply like binomials, replacing \(i^2\) with -1: \((2 + i)(3 - 4i) = 6 - 8i + 3i - 4i^2 = 6 - 5i + 4 = 10 - 5i\).
Conjugates and division
The conjugate of \(a + bi\) is \(a - bi\), and their product \(a^2 + b^2\) is real. To divide, multiply top and bottom by the conjugate of the denominator: \(\frac{5}{1 + 2i} = \frac{5(1 - 2i)}{1 + 4} = 1 - 2i\).
Powers of i
Powers of i repeat in a cycle of four: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), then the pattern repeats. To find \(i^{23}\), divide 23 by 4: remainder 3, so \(i^{23} = -i\).
Complex solutions of quadratics
When \(D < 0\), the quadratic formula gives complex solutions in conjugate pairs. For \(x^2 + 2x + 5 = 0\), \(D = 4 - 20 = -16\), so \(x = \frac{-2 \pm 4i}{2} = -1 \pm 2i\). The fundamental theorem of algebra, proved by Carl Friedrich Gauss in 1799, guarantees every degree-n polynomial has exactly n complex roots, counting repeats.
Worked example
Will the parabola \(y = 2x^2 - 3x + 4\) cross the x-axis? \(D = 9 - 32 = -23 < 0\), so no: since a > 0, it lies entirely above the axis. Its complex roots are \(\frac{3 \pm i\sqrt{23}}{4}\). The discriminant answered the graph question without plotting.
Key takeaways
The discriminant b squared - 4ac tells you the number of real solutions: two if positive, one if zero, none if negative, matching two, one, or no x-intercepts. The imaginary unit i satisfies i squared = -1, so square roots of negatives become multiples of i. Complex numbers a + bi add by parts and multiply like binomials with i squared = -1; divide using the conjugate. Powers of i cycle every four. Negative discriminants give conjugate pairs of complex solutions, and every degree-n polynomial has n complex roots (Gauss, 1799).
Flashcard
b^2 - 4ac
Check your understanding
How many real solutions does \(x^2 - 4x + 4 = 0\) have?