Nature of Roots

The roots of a quadratic equation are the values of x that make the equation equal to zero. A standard quadratic equation is written as ax2 + bx + c = 0, where a  0 and a, b, and c are constants or coefficients. The roots are the values of x that, when substituted into the equation, make the entire expression equal to zero. Depending on the discriminant (D = b2 - 4ac), the quadratic equation can have zero, one, or two real roots.
 

The nature of roots in a quadratic equation helps us identify what kind of solutions we will get when we solve the equation. Every quadratic equation has two roots, which may be real or complex, equal or unequal. The phrase, ‘the nature of roots’ refers to the type of roots the quadratic equation has, whether they are real, complex, rational, irrational, equal, or unequal. 
 

The value of the discriminant determines the nature of the roots of the quadratic equation. The discriminant helps to determine whether the equation has two real roots, one real root, or two complex roots. The number and types of roots that are identified from quadratic equations depend on the discriminant value, which is determined by using the quadratic equation formula given as: x = (-b ± √D)/2a. There are different cases for the number of roots, as shown below:
D > 0
D = 0
D < 0

D > 0
In this case, we get roots that are real and distinct. The quadratic equation has two different real solutions.

D = 0
The roots are real and equal. The quadratic equation has one repeated real solution.

D < 0
The roots are complex and include imaginary numbers, which involve the imaginary number.
 

The discriminant helps to identify the types of roots without solving the equation. The root can be real or complex, and either equal or unequal. 
 

To determine the nature of roots, we can use the quadratic formula to find the solution of a quadratic equation.

                                  ax2 + bx + c = 0
We use the discriminant
                                  x = (-b ± √b2 -4ac)/2a
b2 -4ac is called a discriminant because it is used to find the nature of the roots of a quadratic equation, which is written as x = (-b ± √D)/2a, D = b2-4ac. For example, quadratic equation 2x2 -4x + 3 = 0
Solution:
First step:
   ax2 + bx + c = 0
Identify the values of a, b, and c
a = 2
b = -4
c = 3

Then use the discriminant formula to find the equation
 x = (-b ± √D)/2a
D = b2 - 4ac
D = (-4)2 -4(2)(3)
D = 16 - 24 
D = -8

Use the quadratic formula
x = -b  D2a
x = -(-4)  -8  2(2)
x = 4  -84
The -8 is imaginary because the square root of a negative is not defined within the set of real numbers
x = 4  22i4
x = 2  22i2
The roots are complex, D < 0.
 

To understand the nature of the root of a quadratic equation, we can use the graph.
The graph of a quadratic equation is a parabola, and the points where it intersects the x-axis represent the roots of the equation.
y = ax2 + bx + c
The roots of the equation are the values of x where the graph touches or crosses the x-axis. Let’s see how the graph looks in each case based on the discriminant.

D > 0
The graph crosses the x-axis at two different places, forming a U-shaped curve that touches the x-axis at two points, indicating two distinct real roots.

D = 0
The graph of the quadratic equation touches the x-axis at one point. But it forms a U-shaped parabola with the vertex lying on the x-axis, indicating a single repeated real root..

D < 0
In this case, the graph never touches the x-axis because the equation has no real roots.

The nature of the roots of quadratic equations ax2 + bx + c = 0 depends on the values of the coefficients a, b, and c. Particularly, it is determined by the discriminant D = b2 -4ac. In some cases, the value of b and c significantly affects the nature of the roots of a quadratic equation.

Case 1:
If c = 0
ax2 + bx + c = 0
Then the equation becomes
ax2 + bx + 0= 0
ax2 + bx = 0

This can be factored as:
x(ax + b) = 0

Then set each factor equal to 0:
x = 0
ax + b = 0
x = -ba
So, one root is zero, and the other root is -ba

Case 2:
If b = 0 and c = 0
ax2 + bx + c = 0
ax2 + 0x + 0 = 0
The equation becomes
ax2 = 0
Divide both sides by a 
x2 = 0
The equation has a single real root, and that root is zero.
 

The root is the value of x that makes the equation equal to zero. But in the nature of the roots of a cubic polynomial of degree 3, every cubic polynomial has at least one real root. A cubic polynomial is an equation of the form ax3 + bx2 + cx + d = 0, where a  0, and the equation is of degree 3.
D > 0 
The cubic equation has three distinct real roots.

 D = 0 
The equation has real zeros and at least one repeated zero.
 D < 0 
 The equation has only one real root and two complex conjugate roots.
 

The quadratic equation ax2 + bx + c = 0 has three types of roots based on the value of the discriminant.

• Real and distinct roots
• Real and equal roots
• Complex roots

Real and distinct roots

A quadratic equation can have two different real solutions. The graph of the equation intersects the x-axis at two different points.

Real and equal roots

The equation has only one real root, repeated twice. The graph touches the x-axis at only one point.  D = 0.

Complex roots

The equation has two complex roots and does not touch the x-axis at all in the graph.  D < 0.
 

The nature of the roots of a quadratic equation, whether the roots are real and distinct, real and equal, or complex, plays a vital role in many real-world situations. Here are some examples used in daily life.

• Designing: Engineers use quadratic equations to build the roller coaster, which helps to create the curves that start and end at the ground level.
• Graphics and Editing: In video games, the quadratic equation is used to make the character jump from one place to another. 
• Architecture: Architects use the quadratic equation when constructing arches to accurately model the curved shapes. Using the roots, an architect can determine the points where the arch touches the ground, making it possible to design an arch with a base on the ground.
• Photography: In photography, camera lenses use equations involving roots to focus light and determine focal points. If the roots are complex, they will blur after taking a picture.
• Signal Transmission: In signal transmission, engineers use the quadratic equation to identify the transmission waves.