Quadratic Equations

Algebraic Expressions

Expressions are formed from variables and constants. Terms are added to form expressions. Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively. In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative exponents) is called a polynomial.

Identities

An identity is an equality , which is true for all values of the variables in the equality. Four useful identities are:

  1. (a+b)2 = a2 + 2ab + b2
  2. (a-b)2 = a2 - 2ab + b2
  3. (a+b) (a-b) = a2 - b2
  4. (x+a)(x+b) = x2 + (a+b)x + ab

Quadratic Equations

A quadratic equation in the variable x is of the form

ax2 + bx + c = 0, where a, b, c are real numbers and a≠0.

Roots By Factorization Method

If the quadratic equation ax2 + bx + c = 0 can be expressed in the form (x – a)(x – b) = 0, then the roots of the equation are a and b.

Roots By Quadratic Formula

The roots of a quadratic equation are given by

Nature of Roots

  1. Two distinct real roots, if b2 - 4ac > 0
  2. Two equal roots, if b2 - 4ac = 0
  3. No real roots, if b2 - 4ac < 0

Sum and Product of Roots

If α and β are the roots of the quadratic equation ax2+bx+c=0. Then,

Sum of the roots = α + β = -b/a

Product of the roots = αβ = c/a

If the roots of the quadratic equation are given as α and β, the equation can be written as x2 - x(α + β) + αβ = 0.

Maximum or Minimum Value of a Quadratic Expression

For the quadratic expression ax2 + bx + c, the minimum or minimum value is given by (4ac-b2)/4a and it occurs at -b/2a.

If a > 0, then it will be minimum value. If a < 0, then it will be maximum value.