Class 10 Mathematics · Chapter 4 NotesQuadratic Equations

A quadratic equation in the variable x is an equation of the form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. This is called the standard form. Any equation p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation. A real number α is a root of ax² + bx + c = 0 if aα² + bα + c = 0. The roots of the equation and the zeroes of the polynomial ax² + bx + c are the same. Since a quadratic polynomial has at most two zeroes, a quadratic equation can have at most two roots. Many equations look quadratic but are not, so simplify first and then check the degree.

• Standard formax² + bx + c = 0, a ≠ 0
• α is a root if aα² + bα + c = 0
• A quadratic equation has at most two roots

Quadratic equations appear when we describe real situations in mathematical language. The usual method is to take the unknown quantity as x, use the given condition to build an equation, and then simplify it into standard form. For example, if the breadth of a hall is x metres and its length is one metre more than twice the breadth, the length is (2x + 1) metres. If the carpet area is 300 square metres, then (2x + 1)x = 300, which gives 2x² + x − 300 = 0. Similarly, problems on marbles, toys, ages and speeds can be converted into quadratic equations by writing each condition carefully.

• Take the unknown quantity as x
• Write the given condition as an equation
• Simplify and bring all terms to one side to get ax² + bx + c = 0

If ax² + bx + c can be written as the product of two linear factors, then each factor can be equated to zero to find the roots. This is done by splitting the middle term: find two numbers whose product is ac and whose sum is b, then group the terms and take common factors. For example, 2x² − 5x + 3 = 2x² − 2x − 3x + 3 = (2x − 3)(x − 1), so 2x − 3 = 0 or x − 1 = 0, giving x = 3/2 and x = 1. When the two factors are the same, the equation has two equal roots, as in 3x² − 2√6x + 2 = 0, where both roots are √2/√3.

• Split the middle term using product ac and sum b
• Equate each linear factor to zero
• Repeated factors give two equal roots

Completing the square changes a quadratic equation into a perfect square plus a constant, which can then be solved by taking square roots. The idea is to write x² + bx as (x + b/2)² − (b/2)². This method works for every quadratic equation and is the reasoning behind the quadratic formula. It is especially useful when factorisation is not easy. The steps are: divide by a if a ≠ 1, shift the constant term to the right side, add the square of half the coefficient of x to both sides, write the left side as a perfect square, and then take the square root of both sides.

• Write x² + bx as (x + b/2)² − (b/2)²
• Add the square of half the coefficient of x to both sides
• Take square roots to get the two roots

The roots of ax² + bx + c = 0, when b² − 4ac ≥ 0, are given by the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. To use it, first write the equation in standard form, note the values of a, b and c, and substitute them carefully. The formula comes from solving the general equation by completing the square, so it always gives the roots whenever real roots exist. It is very useful when the coefficients are large or when the equation does not factorise easily. After finding the roots, they can be checked by substituting them back into the original equation.

• Quadratic formulax = (−b ± √(b² − 4ac)) / 2a, provided b² − 4ac ≥ 0
• Identify a, b, c from the standard form before substituting
• The formula works for every quadratic equation with real roots

The expression b² − 4ac is called the discriminant of the quadratic equation ax² + bx + c = 0, because it decides the nature of the roots. If b² − 4ac > 0, the equation has two distinct real roots. If b² − 4ac = 0, the two roots are equal, and each root is −b/2a. If b² − 4ac < 0, there is no real number whose square is b² − 4ac, so the equation has no real roots. The discriminant is therefore a quick test that tells us whether real roots exist and whether they are equal or different, without solving the equation fully.

• b² − 4ac > 0two distinct real roots
• b² − 4ac = 0two equal real roots, each equal to −b/2a
• b² − 4ac < 0no real roots

Quadratic equations are used to solve many practical problems. The method is to read the problem, choose a variable for the unknown, form an equation from the given conditions, solve it, and then check which root makes sense in the situation. For example, if x is a length, breadth, age or speed, it cannot be negative, so a negative root is rejected. In the prayer hall problem, 2x² + x − 300 = 0 gives x = 12 or x = −12.5; since breadth cannot be negative, the breadth is 12 m and the length is 25 m. Problems on consecutive integers, areas, perimeters and production costs are solved in the same way.

• Form the equation from the given condition
• Solve by factorisation or the quadratic formula
• Reject roots that do not fit the real situation