Introduction
A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, c are real numbers and a is not zero. This chapter covers methods to solve quadratic equations, primarily factorisation and the quadratic formula. You will also learn to determine the nature of roots using the discriminant, and apply quadratic equations to solve real-world problems.
Standard Form and Factorisation Method
A quadratic equation in standard form is ax^2 + bx + c = 0. To solve by factorisation, we split the middle term bx into two parts such that their product equals ac, then factorise by grouping. Each linear factor set to zero gives a root of the equation.
Key Points
- •Standard form: ax^2 + bx + c = 0 where a is not zero
- •Split bx into two terms whose coefficients add to b and multiply to ac
- •Factorise by grouping and set each factor equal to zero
- •A quadratic equation has at most two roots
- •Always bring the equation to standard form before solving
Worked Example
Solve: 2x^2 + x - 6 = 0 a = 2, b = 1, c = -6, ac = -12 Two numbers that add to 1 and multiply to -12: 4 and -3 2x^2 + 4x - 3x - 6 = 0 2x(x + 2) - 3(x + 2) = 0 (2x - 3)(x + 2) = 0 x = 3/2 or x = -2
Watch Out
Not all quadratic equations can be factorised easily. If splitting the middle term is difficult, use the quadratic formula directly.
Quadratic Formula
The quadratic formula provides the roots of any quadratic equation ax^2 + bx + c = 0 as x = [-b +/- sqrt(b^2 - 4ac)] / 2a. This formula is derived by completing the square and works for all quadratic equations, whether factorisable or not. The expression under the square root, b^2 - 4ac, is called the discriminant.
Key Points
- •x = [-b +/- sqrt(b^2 - 4ac)] / 2a
- •Works for ALL quadratic equations
- •Remember to identify a, b, c correctly (equation must be in standard form)
- •The +/- gives two roots (which may be equal)
- •Completing the square method is deleted from CBSE 2025-26, but the formula derived from it is required
Worked Example
Solve: x^2 - 4x + 1 = 0 a = 1, b = -4, c = 1 D = b^2 - 4ac = 16 - 4 = 12 x = [4 +/- sqrt(12)] / 2 x = [4 +/- 2*sqrt(3)] / 2 x = 2 +/- sqrt(3) x = 2 + sqrt(3) or x = 2 - sqrt(3)
Watch Out
When the discriminant is not a perfect square, leave the answer in surd form. Do not convert to decimals unless asked.
Nature of Roots (Discriminant)
The discriminant D = b^2 - 4ac determines the nature of roots without actually solving the equation. If D > 0, the equation has two distinct real roots. If D = 0, it has two equal real roots (a repeated root). If D < 0, it has no real roots. The discriminant is a powerful tool for determining conditions under which an equation has real roots.
Key Points
- •D = b^2 - 4ac is the discriminant
- •D > 0: two distinct real roots
- •D = 0: two equal real roots (repeated root)
- •D < 0: no real roots
- •For a quadratic to have real roots, D must be >= 0
Worked Example
For what value of k does 2x^2 + kx + 3 = 0 have equal roots? For equal roots, D = 0 b^2 - 4ac = 0 k^2 - 4(2)(3) = 0 k^2 = 24 k = +/- 2*sqrt(6)
Watch Out
Questions asking 'find k for equal/real/no real roots' are very common in board exams. Set up the discriminant condition and solve.
Word Problems on Quadratic Equations
Quadratic equations arise naturally in many practical situations including area problems, number problems, speed-distance-time problems, and age problems. The key is to translate the word problem into a quadratic equation, solve it, and then reject any solution that does not make sense in the context (e.g., negative lengths or speeds).
Key Points
- •Define the variable clearly
- •Translate the word problem into a quadratic equation
- •Solve using factorisation or quadratic formula
- •Check both roots and reject invalid ones (negative lengths, ages, etc.)
- •Always verify the answer in the original problem context
Worked Example
The product of two consecutive positive integers is 306. Find the integers. Let the integers be x and x + 1. x(x + 1) = 306 x^2 + x - 306 = 0 Using quadratic formula: x = [-1 +/- sqrt(1 + 1224)] / 2 = [-1 +/- 35] / 2 x = 17 or x = -18 Since we need positive integers, x = 17. The integers are 17 and 18. (Check: 17 x 18 = 306)
Quick Summary
- ✓Standard form of quadratic equation: ax^2 + bx + c = 0 (a != 0)
- ✓Factorisation: split middle term, factorise by grouping
- ✓Quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / 2a
- ✓Discriminant D = b^2 - 4ac determines nature of roots
- ✓D > 0: two distinct real roots; D = 0: equal roots; D < 0: no real roots
- ✓In word problems, always reject roots that do not fit the context
- ✓For equal roots: set D = 0; for real roots: set D >= 0
Key Formulas
Quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / 2a
Discriminant: D = b^2 - 4ac
D > 0: two distinct real roots
D = 0: two equal real roots
D < 0: no real roots
Sum of roots = -b/a, Product of roots = c/a
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