Nature of Roots of a Quadratic Equation

Quadratic Equations is an essential part of Algebra in Mathematics. Quadratic Equations are the equations that contain at least one term that is in square form or squared.  Quadratic equations are also known as the equation of second degree. The standard form of Quadratic equation ax² + bx + c = 0 with a, b and c being constants, or numerical coefficients or real numbers and they are not equal to 0, and x being an unknown variable whose value will be discovered by solving the equation by using varied quadratic equation methods and the two powers of x means square on x where the nature of roots can be real or either estimated or assumed.

How are Quadratic equations different from Quadratic Polynomials? 

A quadratic polynomial becomes a quadratic equation when the polynomial is equated to 0.  For Example,  a Quadratic Polynomial is ax + bx² + c while  a Quadratic Equation is  ax + bx² + c =0. 

Methods used for finding roots of Quadratic Equations 

The roots of the Quadratic equation is the value of an unknown factor of the equation. For Example, if ax + bx² + c =0 then the root of the quadratic equation will be the value of x. Following  are some methods that can be used for finding roots of Quadratic Equations: 

• Factorization method

• Quadratic Formula method

• Completing the square method.

Graphical Representation of Quadratic Equation 

Quadratic Equations can also be represented on the graphs too and the representation of the quadratic equations on the graph is known as a graphical representation of the quadratic equations. The roots of the quadratic equation are the points that touch the x-axis. For example, the roots of x² + 9 will be 3 and -3. 

Following are some important  points that one should keep in mind during graphical representation of the quadratic equations : 

• If in the graph of a quadratic equation, no point touches the x-axis then the following equation does not have any roots. 

• If the graph of the quadratic polynomial equation touches the x-axis at two different points, then it implies that the equations have real and distinct roots.

• If the graph of the quadratic polynomial equation touches the x-axis, then it implies that it has real and equal roots.

Type of Roots 

There are three types of roots : 

• Complex roots

• Real and equal roots

• Real and Distinct roots 

Nature of Roots 

The discriminant is the expression under the square root. The discriminant is represented as Δ. Δ is the Greek symbol for the letter D.

The discriminant determines the nature of the roots of a quadratic equation. The word ‘nature’ refers to the types of numbers the roots can be like namely real, rational, irrational or imaginary etc. 

• If Δ<0 then the roots are non-real and in graphical representation, the curve does not intersect the x-axis at any point. 

• If Δ=0 then the roots are real and equal and in graphical representation, the curve intersects at only one point at the x-axis. 

• If Δ> 0 then roots are either rational or irrational and in graphical representation, the curve intersects at two distinct points at the x-axis. 

• Δ is the square of a rational number then the roots are rational.

• Δ is not the square of a rational number then the roots are irrational and can be expressed in decimal form.

Solved Examples

1. Discuss the nature of the roots of a quadratic equation 2x2 – 8x +3 = 0

Solutions: Here, the coefficients are all rational. The discriminant D for a given equation will be

D = b2 – 4ac = (-8)2- 4*2*3

=64-24

= 40 > 0

We can see, the discriminant of the given quadratic equation is positive but not a perfect square.

Hence, the roots of a quadratic equation are real, unequal, and irrational.

2. Determine the value of k for which the quadratic expression (x-k) (x-10) +1 =0 has integral roots.

Solution : The given quadratic equation can be rewritten as 

x2 – (10 + k) x +1 + 10k = 0

b2 – 4ac = 100 + k2 + 20k – 40k = k2 -100k  + 96 = (k - 10)2 - 4

The quadratic equation will have an integral root if the value of discriminant > 0, D is a perfect square, a=1 and b and c are considered as integers.

i.e. (k-10)2 - D =4

Since the discriminant is a perfect square. Hence, the difference of the two perfect squares in R.H.S will be 4 only when D=0 and  (k-10)2 =4

→ k-10 = ± 2, Hence, the values of k = 8 and 12.

Facts 

• Sudhara, the Sanskrit philosopher, introduced the formula to find the two roots of a quadratic equation.

• The Babylonians introduced a technique known as completing the square to resolve common issues with areas by 400 BC.

• The first reliable mathematical try to introduce quadratic formula was done by Pythagoras in 500 BC.