Nature of Roots

A degree two polynomial equation is called a quadratic equation. The basic form of a quadratic equation can be represented as ax²+ bx+c=0, where a, b and c are considered to be real numbers and the coefficients of the variable x².ie.,  a is not equal to 0. A quadratic equation will invariably has two roots. Some of the methods of determining the roots of a quadratic equation are:

• Factorisation
• Completing Square
• Quadratic formula

Only quadratic Equations having real roots can be factorised. In graphical sense, roots can defined as the points where the graph of the quadratic equation intersects the x-axis.

Quadratic Formula and The Discriminant

We mainly use the Quadratic Formula to determine the nature root of Quadratic equation. Now for any given quadratic equation  ax²+ bx+c=0, the quadratic formula is given as:,br>

x=-b±√b²-4ac/2a

Here, the expression (b²-4ac) is called the discriminant. Depending on the value of the discriminant, the nature of the root is determined. It is represented as D. The following cases are formed:

• CASE1: If D=0, which means b²-4ac=0, the roots are said to be real and equal.

Now we have been given that b²-4ac=0, substituting in quadratic formula, we get

x=-b+√02a,-b-√02

x=-b2a,-b2a

Hence, the roots are real, rational and equal.

• CASE2: If D>0, which means  b²-4ac>0 and a perfect square, the roots are said to be real, unequal and rational.

We take the quadratic formula and substitute the values, we get

x=-b+√D2a,-b-√D2a

Hence, the roots are distinct, real and rational.

• CASE3: If D>0, which means b²-4ac>0 but not a perfect square, the roots are real, unequal but irrational.
• CASE4: If D<0, which means b²-4ac<0 the roots are said to be imaginary.
• CASE5: If D≥0, which means b²-4ac ≥ 0, the roots are real. After combining results from case1, case2 and case3, the real part is common. Hence, the roots in this case are real.

Graphical Significance of Nature of Roots

Factorization is feasible for any quadratic equations with real roots.The quadratic equation on the graph is represented as a parabola. The roots have physical relevance since the graph of an equation touches the x-axis at its roots. In the Cartesian plane, the x-axis exemplifies the real line. Therefore, if an equation has unreal roots, it will not intersect the x-axis and hence cannot be factored.

Solved Examples

Example1 Find the roots of the Quadratic equation x²+ 9x+14=0 and also determine the nature of the roots.

Solution

On the comparison between the given equation with ax²+ bx+c=0, a=1, b=9 and c=14.

Now we calculate the discriminant, which implies

D = b²-4ac

= 9²-4(1)(14)

= 81 – 56

D = 25 > 0

Substituting the value of the discriminant in the Quadratic formula to get the roots, we get

x=-b+√D/2a,-b-√D/2a

x=-9+√25/2(1),-9-√25/2(1)

x=-9+5/2,-9-5/2

x=-4/2,-14/2

x   = -2, -7

Therefore, roots of the given equation are -2 and -7, which are non-equal, real and distinct, since D>0.

Example2  For what value of m, are the roots of the equation (3m+1)x²+ (11+m)x+9=0 equal?

Sol.    With the comparison of the given equation with ax²+ bx+c=0 ;

we get : a = 3m + 1, b = 11 + m and c = 9

∴ Discriminant, D = b²-4ac

= (11+m)²-4(3m+1)(9)

=   121+22m + m²-108m-36

=   m²-86m +85

=  m²-85m-m +85

= m(m-85)-1(m-85)

= (m-85)(m-1)

Since the roots are equal, D = 0

⇒ (m – 85) (m – 1) = 0

⇒ m – 85 = 0 or m – 1 = 0

⇒ m = 85 or m = 1

Conclusion

We learned to find the nature of roots of quadratic equations using the discriminant formula, how the roots are calculated using different methods. The quadratic formula is used essentially to determine the nature of the roots. The nature described by roots of a quadratic equation are mainly real roots, unreal roots, irrational roots, and imaginary roots.