Quadratic Equation

Quadratic Equation

Quadratic Equation

A quadratic equation is a second degree equation in a single variable. For example: x2 – 25 = 0, 4x2 – 81 = 0, x2 + 3x + 2 = 0, 2x2 – 7x – 15 = 0 etc.

The general form of quadratic equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0 is called the standard form of a quadratic equation.

Pure quadratic equation

A quadratic equation of the form ax2 + c = 0, where the middle term with the variable containing power 1 is missing, is known as a pure quadratic equation. For example: x2 – 9 = 0, 2x2 – 5 = 0 etc. are pure quadratic equations.

Adfected quadratic equation

A quadratic equation where the second power of variable as well as the first power of the same variable occurs is called the adfected quadratic equation. For examples: x2 + 9x + 14 = 0, 2x2 + 5x – 3 = 0 etc. are the adfected quadratic equations.

Solution of quadratic equations

A quadratic equation is a second degree equation. So we obtain two solutions (or roots) of the variable from a quadratic equation. In other words, a quadratic equation has two roots. We can solve a quadratic equation by following three methods:
i.         Factorization Method
ii.       Completing the Square Method
iii.      By using formula

Solving Quadratic Equation by Factorization

While solving quadratic equation by factorization method, we factorize the second degree polynomial ax2 + bx + c and expressed as the product of two linear factors. Then each linear factor is separately solved to get the required solutions of the equation by applying zero factor property.

In zero factor property, if p.q = 0, then either p = 0 or q = 0. In other words, if the product of two numbers is 0, then either one or both of the number must be 0.

We use the following steps to solve a quadratic equation by the factorization method.

Steps:

i.       Remove the brackets of fractions if there is any in the equation.
ii.      Simplify and reduce the equation in the form of ax2 + bx + c = 0.
iii.     Factorize the left hand side into two factors.
iv.     Use the above property of numbers and write each factor equal to 0.
v.      Solve both factors and get two values of the variable.

The values of the variable are also called the roots of the equation.

This process of factorization Method will be clear by the following worked out examples.

Workout Examples

Example 1: Solve the following quadratic equations by factorization method:
a.    x2 – 9 = 0
b.    2x2 = 8
c.     x2 – 3x – 28 = 0

Solution:  
    a.     x2 – 9 = 0
            or,    x2 – 32 = 0
            or,    (x + 3)(x – 3) = 0

                  Either, x + 3 = 0  or, x = - 3

                    Or, x – 3 = 0  
                     or, x = 3

             x = 3, –3

             
    b.     2x2 = 8
            or,    x2 = 8/2
            or,    x2 = 4
            or,    x2 4 = 0
            or,    x2 22 = 0
            or,    (x + 2)(x – 2) = 0

                  Either, x + 2 = 0  or, x = - 2

                    Or, x – 2 = 0  
                     or, x = 2

              x = 2, –2

           
            c.     x2 – 3x – 28 = 0
            or,    x2 – (7 – 4)x – 28 = 0
            or,    x2 – 7x + 4x – 28 = 0
            or,    x(x – 7)+4(x 7) = 0
            or,    (x – 7)(x + 4) = 0

                  Either, x 7 = 0  or, x = 7

                        Or, x + 4 = 0  
                         or, x = 4

              x = 7, –4


Example 2: Solve:   (x+1)/(x-1)  –  (x-1)/(x+1)  =  5/6 by factorization method.
or,        5(x2 – 1) = 24x
or,        5x2 – 5 = 24x
or,        5x2 – 24x – 5 = 0
or,        5x2 – (25 – 1)x – 5 = 0
or,        5x2 – 25x + x – 5 = 0
or,        5x(x – 5) + 1(x – 5) = 0
or,        (x – 5)(5x + 1) = 0

          ∴ Either, x 5 = 0  or, x = 5
            Or, 5x + 1 = 0  or, 5x = 1 or x = 1/5

             x = 5, –1/5


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