Quadratic Equations


In this section, we will learn about quadratic equations, their concepts, roots, and graphs. 

Quadratic
 Equation: It is an equation of the form  ax^{2}+bx+c= 0 . Where a, b, c are real numbers , a ≠ 0 and x is the variable.
We can also say that it is a polynomial equation of degree 2 with a single variable.

Standard form of a quadratic equation.
ax^{2}+bx+c= 0 and a ≠ 0  

Examples:

  1. x^{2}+2x+2= 0
    Here a= 1, b = 2 and c=2.
  2. x^{2}+3x-1= 0
    Here a= 1, b = 3 and c = – 1.
  3. 4x^{2}-8x= 0
    Here a= 4, b = -8 and c = 0.
  4. 2x^{2}-1= 0
    Here a= 2, b = 0 and c = -1.

Question: Check whether the following equations are quadratic or not.
i) x^{2}+5x+6= 0
Answer: Yes, the above equation is a quadratic equation. It is of the form ax^{2}+bx+c= 0 where a= 1, b = 5 and c = 6.
ii)x^{2}+2= 0
Answer: Yes, the above equation is a quadratic equation. It is of the form ax^{2}+bx+c= 0 where a= 1, b = 0 and c = 2. 
iii) x-2= 0
Answer: No, the above equation is not a quadratic equation because it is not of the form ax^{2}+bx+c= 0. As here a = 0. For a quadratic equation  a ≠ 0.

Methods of finding the roots of quadratic equations

When we put x = α in the quadratic equation of the form ax^{2}+bx+c= 0 ,  such that aα ^{2}+bα +c= 0 i.e α  satisfies the equation. Thus, α is called the root or solution of quadratic equation.

  1. By Quadratic Formula

    The roots of quadratic equation ax^{2}+bx+c= 0 , a 0 can be given by,

    x = \frac{-b\pm \sqrt{b^{2}-4ac}}{2a} 

    provided b^{2}-4ac ≥ 0.

    Question : Find the roots of the quadratic equation 3x^{2}- 5x + 2= 0 by quadratic formula.
    Solution: Given, 3x^{2}- 5x + 2= 0 
    Here a = 3 , b = -5 and c= 2
    By quadratic formula

    x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}

    Now b^{2}-4ac = (-5)^{2} – 4.2.3 = 25 – 24 = 1

    x=\frac{-(-5)\pm \sqrt{1}}{2.3}

    ⇒ x=\frac{5\pm 1}{6}

    ⇒ x=1 or x=\frac{2}{3}

    Thus. the roots are 1 and \frac{2}{3}
  2. Finding roots by Factorisation:

    This can be done by splitting the middle term.
    Let us understand this by an example.
      i) x^{2}-3x-10= 0
    This can written as 
    x^{2}-5x+2x-10= 0
    x ( x – 5) + 2 ( x – 5) = 0
    ( x -5 ) (x + 2) = 0
    Thus, x = – 2 and x = 5
    Here we found the roots of the equation by factorising the equation into two linear factors and equating each factor to zero.

Nature of roots and Discriminant

We know the roots of quadratic equation ax^{2}+bx+c= 0 , a 0 can be given by,

x = \frac{-b\pm \sqrt{b^{2}-4ac}}{2a} 

It is to be noted that b^{2}-4ac determines the nature of the roots. Also,  b^{2} – 4ac is called the discriminant of quadratic equation. 

S.No Discriminant Nature of roots
1 b^{2}-4ac> 0 Two distinct and real roots
2 b^{2}-4ac = 0 Two equal and real roots
3 b^{2}-4ac< 0 No real roots

Graph of Quadratic Equations

The graph of a quadratic Equation of the form ax^{2}+bx+c= 0 where  a ≠ 0 is a PARABOLA. 
The
coefficient “a” determines whether the parabola will be Upwards or Downwards.

  •  If a > 0, it will be an UPWARD PARABOLA 
  •  If a < 0, it will be a DOWNWARD PARABOLA 

 

Quadratic equations graph

Arithmetic Progression
Compound Interest
Simple Interest