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MathidiaMathematics Academy
Proof and advanced algebra
Lesson 3 of 75
Standard practice

Quadratics and discriminants

Objective. Choose efficient quadratic methods and reason from the discriminant.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A quadratic ax²+bx+c=0 can be solved by factorising, completing the square, or the formula x=(−b±√(b²−4ac))/(2a), and the discriminant b²−4ac tells you how many real roots exist without solving fully. A positive discriminant gives two distinct real roots, zero gives one repeated root, and a negative discriminant gives no real roots. For 2x²−4x+5=0, b²−4ac=16−40=−24<0, so there are no real solutions and the graph never crosses the x-axis. Choosing completing the square is efficient when you also need the turning point.

Worked example

Use the discriminant to determine how many real roots the quadratic 3x²+2x+5=0 has.

3 … 2
Illustrated card showing 3 … 2
  1. 1Identify a=3, b=2, c=5
  2. 2Compute the discriminant b²−4ac = 4 − 60
  3. 3Simplify to get −56, which is negative
  4. 4A negative discriminant means no real roots

Answer: No real roots

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Common mistake. Students forget the ± sign in the quadratic formula and give only one root, e.g. solving x²−x−6=0 and stating x=3 while missing x=−2; always write both roots from ±√(discriminant).

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