Solution of Quadratic Equation

A zero of a polynomial is that real number, which when substituted for the variable makes the value of the polynomial zero. In case of a quadratic equation, the value of the variable for which LHS and RHS of the equation become equal is called a root or solution of the quadratic equation.

If α is a zero of a polynomial p(x), then (x - α) is a factor of p(x) and conversely, if (x - α) is a factor of a polynomial, then α is a zero of the polynomial.

There are two algebraic methods for finding the solution of a quadratic equation:

  1. Factor Method
  2. Using the Quadratic Formula

Factor Method

Example: Solve the equation 6x2 + 7x - 3 = 0 by factorisation.

6x2 + 7x - 3 = 0

By breaking middle term, we get

6x2 + 9x - 2x - 3 = 0

3x(2x + 3) - 1(2x + 3) = 0

(2x + 3)(3x - 1) = 0

This gives 2x + 3 = 0 or 3x - 1 = 0

x = -3/2 or x = 1/3

Example: Solve x2 + 2x + 1 = 0

x2 + 2x + 1 = 0

(x + 1)2 = 0

x + 1 = 0

x = -1

Therefore, x = -1 is the only solution. Here, the equation has two solutions and these are coincident.

Quadratic Formula

ax2 + bx + c = 0

Multiplying both sides by 4a to make the coefficient of x2 a perfect square, of an even number, we get

4a2x2 + 4abx + 4ac = 0

(2ax)2 + 2(2ax)b + (b)2 + 4ac = b2

(2ax)2 + 2(2ax)b + (b)2 = b2 - 4ac

(2ax + b)2 = b2 - 4ac

$$ 2ax + b = \pm \sqrt{b^2 - 4ac} $$

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

This gives two solutions of the quadratic equation.

Discriminant

Here, the expression (b2 - 4ac), denoted by D, is called Discriminant, because it determines the number of solutions or nature of roots of a quadratic equation.

Nature of Roots

For a quadratic equation ax2 + bx + c = 0, a ≠ 0, if

  1. D = b2 - 4ac > 0, the equation has two real distinct roots.
  2. D = b2 - 4ac = 0, then equation has two real equal roots.
  3. D = b2 - 4ac < 0, the equation will not have any real root, since square root of a negative real number is not a real number.

Thus, a quadratic equation will have at the most two roots.