Quadratic Formula

Quadratic Formula

Quadratic Formula

A formula to calculate the roots or solutions of a quadratic equation of the form ax2 + bx + c = 0 is known as the Quadratic Formula which is given as,         
Quadratic Formula
where, a is coefficient of x2, b is coefficient of x and c is constant term. We use this formula to find the required solutions (roots) of the given quadratic equation.

Some special conditions:
1.    When b2 > 4ac, there are two distinct real roots.
2.    When b2 = 4ac, there is a single real root.
3.    When b2 < 4ac, there is no real roots.

Derivation of Quadratic Formula

Consider the general quadratic equation
            ax2 + bx + c = 0, where a, b and c are constant values and a≠ 0.
Rewriting the given equation as
            ax2 + bx = – c
Dividing both sides by the coefficient of x2 i.e. by a we get
Derivation of Quadratic Formula
Hence the Quadratic Formula is derived.

Workout Examples

Example 1: Solve x2 – 7x + 12 = 0 by using quadratic formula.

Solution: Here,
                        x2 – 7x + 12 = 0
                Here,
                        a = 1
                        b = –7
                        c = 12
            Using quadratic formula,
Example 1: Solving x^2 – 7x + 12 = 0 by using quadratic formula

              x = 4, 3


Example 2: Solve (p – q)x2 + (p + q)x + 2q = 0 by using quadratic formula.

Solution: Here,
                        (p – q)x2 + (p + q)x + 2q = 0
                Here,
                        a = (p – q)
                        b = (p + q)
                        c = 2q
            Using quadratic formula,
Example 2: Solving (p – q)x^2 + (p + q)x + 2q = 0 by using quadratic formula


Example 3: Solve 1/(x-2)+2/(x-1)= 6/x by using quadratic formula.
or,        6x2-18x+12 = 3x2-5x
or,        6x2–18x+12–3x2+5x = 0
or,        3x2–13x+12= 0

                Here,
                        a = 3
                        b = –13
                        c = 12
            Using quadratic formula,
applying quadratic formula


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