Methods of Solving Quadratic Equations

Introduction to methods of solving quadratic equations:

The term "quadratic" comes from quadratures, which is the Latin word for "square" Quadratic equations can be solved by factoring,completing the square,graphing are used in the Quadratic formula.

In general quadratic equation is a polynomial equation of the second degree and Generally it defind as the  ax^2+bx+c=0 in which 'x'a,b,c are the constants with a not equal to zero and the value a=0 then it is a linear equation.The constants a, b, and c, are called respectively, the quadratic Coefficient the linear coefficient and the constant terms or free term. can represents a variable and

A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real

The roots are given by the quadratic formula

x= -b`+-` √ (b2-4ac)/2a 

Steps for solving quadratic equation :

Step 1: Put all terms on one side leaving zero on other side.

Step 2: Factorize.

Step 3: Use the theorem and set each factor to zero.

Step 4: Solve each equation.


Methods for Solving Quadratic Equations :-

Method 1: FACTORING

Method 2:PRINCIPLE OF SQUARE ROOTS.

Method 3:COMPLETING THE SQUARE

Method 4:QUADRATIC FORMULA

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Factoring Method:

Set the equation equal to zero. If the quadratic side is factorable, factor, then set each factor equal to zero.
Example: x2 = −5x − 6
Move all terms to one side x2 + 5x + 6 = 0
Factor (x + 3)(x + 2) = 0
Set each factor to zero and solve x + 3 = 0 x + 2 = 0
x = −3 x = −2

Principle of Square Roots Method:

If the quadratic equation involves a SQUARE and a CONSTANT (no first degree term), position the square on one side and the constant on the other side. Then take the square root of both sides.

Example 1: x2 −16 = 0
Move the constant to the right side x2 = 16
Take the square root of both sides x = ± 4, which means x = 4 and x = −4

Completing the Square Method:

If the quadratic equation is of the form ax2 + bx + c = 0, where a ≠ 0 and the quadratic expression is
not factorable, try completing the square.
Example: (x – 4)2 = 5
x – 4 = ± √(5)
x = 4 ± √(5)
x = 4 – √(5)  and  x = 4 + √(5)

Quadratic Formula Method:

Any quadratic equation of the form ax2 + bx + c = 0, where a ≠ 0 can be solved for both real and
imaginary solutions using the quadratic formula:

x= -b`+-` √ (b2-4ac)/2a