Solving quadratics and reading the nature of their roots
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. It can be solved by splitting the middle term and factorising, by completing the square, or by the quadratic formula x = [−b ± √(b² − 4ac)] / 2a. For x² − 5x + 6 = 0, splitting −5x as −2x − 3x gives (x − 2)(x − 3) = 0, so x = 2 or x = 3. The formula always works, even when factors are hard to spot.
The discriminant D = b² − 4ac decides the nature of the roots. If D > 0 there are two distinct real roots, if D = 0 there are two equal real roots, and if D < 0 there are no real roots; in Class 11, a negative discriminant gives a pair of complex conjugate roots. The sum of the roots is −b/a and their product is c/a, so the quadratic with roots α and β is x² − (α + β)x + αβ = 0. Word problems on areas, speeds and ages often lead to quadratics, and one root may have to be rejected because it is negative or unrealistic.
Why it matters
Quadratics describe the path of a thrown ball, the area of a field with an unknown side and the profit of a business that depends on its price. They are a regular part of Class 10 board papers, return in Class 11 with complex roots, and are central to entrance exams. Recognising a quadratic hidden inside a word problem is a skill used throughout physics and economics.
Key terms
- Discriminant
- The value b² − 4ac, which tells whether the roots are real and distinct, equal or not real.
- Quadratic formula
- The general solution x = [−b ± √(b² − 4ac)] / 2a for any quadratic equation.
- Splitting the middle term
- Rewriting bx as two terms whose coefficients multiply to ac, so the expression can be factorised.
- Real roots
- Solutions of the equation that are real numbers and can be plotted on the number line.