Methods to solve the Quadratic Equations

There are three methods to solve the Quadratic Equations-

  1. Factorization Method

In this method, we factorize the equation into two linear factors and equate each factor to zero to find the roots of the given equation.

Step 1: Given Quadratic Equation in the form of [mathtype]a{x^2} + bx + c = 0  .

Step 2: Split the middle term bx as mx + nx so that the sum of m and n is equal to b and the product of m and n is equal to ac.

Step 3: By factorization we get the two linear factors (x + p) and (x + q)

[mathtype]a{x^2} + bx + c = 0  = (x + p) (x + q) = 0

Step 4: Now we have to equate each factor to zero to find the value of x.

 

[mathtype]\begin{array}{l}{x^2} - 2x - 15 = 0\\(x + 3)(x - 5) = 0\\x + 3 = 0\& x - 5 = 0\\x =  - 3\& x = 5\\x = \left\{ { - 3,5} \right\}\end{array} 

These values of x are the two roots of the given Quadratic Equation.

  1. Completing the square method

In this method, we convert the equation in the square form [mathtype]{(x + a)^2} - {b^2} = 0  to find the roots.

Step1: Given Quadratic Equation in the standard form [mathtype]a{x^2} + bx + c = 0  .

Step 2: Divide both sides by a

[mathtype]{x^2} + \dfrac{b}{a}x + \dfrac{c}{a} = 0 

Step 3: Transfer the constant on RHS then add square of the half of the coefficient of x i.e. [mathtype]{\left( {\dfrac{b}{{2a}}} \right)^2}  on both sides

[mathtype]{x^2} + \dfrac{b}{a}x + \dfrac{c}{a} = 0 

[mathtype]{x^2} + 2\left( {\dfrac{b}{{2a}}} \right)x + {\left( {\dfrac{b}{{2a}}} \right)^2} =  - \dfrac{c}{a} + {\left( {\dfrac{b}{{2a}}} \right)^2} 

Step 4: Now write LHS as perfect square and simplify the RHS.

[mathtype]{\left( {x + \dfrac{b}{{2a}}} \right)^2} = \dfrac{{{b^2} - 4ac}}{{4{a^2}}} 

Step 5: Take the square root on both the sides.

[mathtype]x + \dfrac{b}{{2a}} =  \pm \sqrt {\dfrac{{{b^2} - 4ac}}{{4{a^2}}}}  

Step 6: Now shift all the constant terms to the RHS and we can calculate the value of x as there is no variable at the RHS.

[mathtype]x =  \pm \sqrt {\dfrac{{{b^2} - 4ac}}{{4{a^2}}}}  - \dfrac{b}{{2a}} 

  1. Quadratic formula method

In this method, we can find the roots by using quadratic formula. The quadratic formula is

[mathtype]x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} 

where a, b and c are the real numbers and b– 4ac is called discriminant.

To find the roots of the equation, put the value of a, b and c in the quadratic formula.

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