Discriminant

Statement

The discriminant is a key part of the quadratic formula and helps determine the nature of the solutions of a quadratic equation of the form \(ax^2 + bx + c = 0\). It is given by:

\[\Delta = b^2 - 4ac\]

The value of \(\Delta\) tells us how many and what type of solutions (roots) the quadratic equation has.

Why it’s true

• The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• The expression inside the square root, \(b^2 - 4ac\), is the discriminant.
• If the discriminant is positive, the square root is a real number, giving two distinct real roots.
• If it is zero, the square root vanishes, giving one repeated real root.
• If it is negative, the square root is imaginary, giving two complex conjugate roots.

Recipe (how to use it)

1. Identify coefficients \(a\), \(b\), and \(c\) from the quadratic equation \(ax^2 + bx + c = 0\).
2. Calculate \(\Delta = b^2 - 4ac\).
3. Interpret the result:
   • \(\Delta > 0\): two real and distinct roots.
   • \(\Delta = 0\): one real repeated root.
   • \(\Delta < 0\): two complex roots.

Spotting it

The discriminant is relevant whenever you are asked about the number or type of solutions of a quadratic, without needing to solve it fully.

Common pairings

• Quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\).
• Completing the square (where the discriminant shows whether the square term balances).
• Graph of a parabola (the discriminant tells how many times it cuts the x-axis).

Mini examples

1. Given: \(x^2 - 5x + 6 = 0\). Find: discriminant. Answer: \(b^2 - 4ac = 25 - 24 = 1 > 0\), so two real roots.
2. Given: \(x^2 + 4x + 4 = 0\). Find: discriminant. Answer: \(16 - 16 = 0\), one repeated root.
3. Given: \(x^2 + 2x + 5 = 0\). Find: discriminant. Answer: \(4 - 20 = -16 < 0\), two complex roots.

Pitfalls

• Forgetting to square \(b\) correctly.
• Using the wrong signs for \(a\), \(b\), or \(c\).
• Mixing up the discriminant with the quadratic formula itself.

Exam strategy

• Quickly calculate \(\Delta\) before deciding how to solve the quadratic.
• If \(\Delta\) is negative, don’t waste time looking for real roots.
• If \(\Delta\) is a perfect square, expect rational roots.

Summary

The discriminant \(\Delta = b^2 - 4ac\) is a shortcut to determine the nature of quadratic roots. It helps decide whether solutions are real or complex, distinct or repeated, and links algebra with the graph of the parabola.