Roots of a Quadratic Equation
A quadratic equation is a polynomial equation of the second degree. In other words, a second degree polynomial in x equated to 0, where coefficient of x^2 =/=0 will be a quadratic equation.
The general form is
ax^2 + bx + c= 0 , where x represents a variable or an unknown, and a, b, and c are constants with a ≠ 0. (If a = 0, the equation is a linear equation.)
Roots of a quadratic equation can be found out by two ways
a) by factorising the expression on the LHS of a quadratic equation.
b) by using the quadratic formula
A) Finding the roots by factorising the LHS of quadratic equation
If we can write the quadratic equation ax^2 + bx + c= 0 in the form of (x-p)(x-q)=0 then p and q will be the roots.
Method: Step by Step
1.) Write down the coefficient of x^2, x and constant term i.e.a,b and c respectively.
2.) Write b as the sum of 2 quantities such that their product is equal to ac.
3.) Now rewrite the bx term split in the 2 quantities found above.
4.) Take the common factor from 1st 2 terms and the last 2 terms.
5.) Rewrite the entire LHS such that (x-p)(x-q)= 0 . So p and q are roots of quadratic equation
Example: x^2 – 4x – 5 = 0 is a quadratic equation find its roots.?
Ans. x^2 – 4x – 5 = 0
a=1, b=-4, c= -5 so ac = -5 –> so if we split b=-4 as -5 + 1 we get -5×1=-5 = ac
–> x^2 – 5x+ x – 5 =0 –> x(x-5)+1(x-5)= 0 –>(x+1)(x-5)=0 –>(x-(-1))(x-5)= 0 –> x=-1,5 are the roots of the equation
B) Using the quadratic formula
The standard formula for finding the roots of a quadratic equation ax^2+bx+c= 0 is x=[-b+-sqrt(b^2-4ac)]/2a.
where D= b^2-4ac is called Discriminant of the quadratic equation
so lets solve the above quadratic equation by formula.
a=1, b=-4, c=-5. so b^2-4ac = (-4)^2 – 4x1x(-5) = 16 +20 =36
so x=[-(-4)+-sqrt(36)]/(2.1) = (4+-6)/2 = 5, -1
The following calculator(http://www.math.com/students/calculators/source/quadratic.htm) by Maths.com can be used to verify your answer while calculating roots of quadratic equation.
Discriminant helps in determining the nature of the roots quadratic equation (whether it is real or complex) without actually calculating the roots of the equation. We will discuss the Nature of the roots of quadratic equation in next article.
