Discriminant

A quadratic equation is written in the form ax² + bx + c = 0, where a, b, and c are coefficients, and x is the variable. The roots are the values of x that make the equation true. There are different methods to find the roots of quadratic equations, such as:

 

• Factoring
• Completing the square 
• The quadratic formula.

In mathematics, the discriminant is a value calculated from the coefficients, used to determine the nature of the roots. It is denoted by Δ or D. The value of the discriminant can be positive, negative, or zero.  

• When finding the roots of a quadratic equation, the discriminant plays a major role. 

• The nature of the roots are real and distinct, real and equal or complex, is determined by the value of the discriminant.

• For a quadratic equation, \(ax^2 + bx + c = 0\), the value of the discriminant depends on the coefficients a, b, and c, and it is used to understand the type of solutions. 
   

The nature of the roots in a quadratic equation is determined by the value of the discriminant. For a quadratic equation \(ax^2 + bx + c = 0\), the discriminant formula is \(D = b^2 - 4ac\). 


The discriminant formula for a cubic equation \(ax^3 + bx^2 + cx + d = 0, \quad D = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2\)

As we know, there is a formula to find the value of the discriminant, so by substituting the values in the formula, we find the discriminant. In this section, we will learn how to find the discriminant of a quadratic and cubic equation. 

For a quadratic equation \(ax^2 + bx + c = 0\), the value of the discriminant is \(D = b^2 - 4ac\).  

We know that for a quadratic equation \(ax^2 + bx + c = 0\), the value of \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

As \(D = b^2 - 4ac\), the value of x can be written as:

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

For example, finding the discriminant of a quadratic equation \(3x^2 - 4x + 8 = 0\)


The formula to find the discriminant is \(D = b^2 - 4ac\)


Here, a = 3, b = -4, and c = 8

 

\(D = (-4)^2 - 4 \times 3 \times 8\)


= 16 - 96 


= -80. 
 

For a cubic equation \(ax^3 + bx^2 + cx + d = 0\) the discriminant can be calculated using the formula:

\(D = 18abcd - 4b^3d + b^2c^2 - 4ac - 27a^2d^2\)

For example, find the discriminant of the cubic equation: \(x^3 - 6x^2 + 11x - 6 = 0\)


To find the discriminant, we use the formula: \(D = b^2 c^2 - 4 a c^3 - 4 b^3 d - 27 a^2 d^2 + 18 a b c d\)


Here, a = 1
b = -6
c = 11
d = -6

\(D = (-6)^2 (11)^2 - 4(1)(11)^3 - 4(-6)^3(-6) - 27(1)^2(-6)^2 + 18(1)(-6)(11)(-6)\)

\(D = 4356 -5324 = -968; -968 - 5184\)

\(D = -6152; -6152 - 972 = -7124 \)

\(D = -7124 + 7128\)

D = 4 

The roots of a quadratic equation are calculated using the formula:

  

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

The quadratic equation can be written as: 

\(x = \frac{-b \pm \sqrt{D}}{2a}, \quad \text{where } D = b^2 - 4ac\)

The discriminant is used for the nature of the roots, which means what type of numbers the root can be. A root can be real, rational, irrational, or imaginary. 

•  If the Discriminant is Positive: If the discriminant is positive, D > 0, then there are two real roots for the quadratic equation. In other words, if D > 0, x becomes x = (-b ± √positive number) / 2a, as the square root of a positive number is always a real number. 

• If the Discriminant is Negative: When the discriminant of a quadratic equation is negative (D < 0), then it has two complex conjugate roots. If D < 0, the value of x = (-b ± √negative number) / 2a. The square root of a negative number always results in an imaginary number.

• If the Discriminant is Equal to Zero: If the discriminant of a quadratic equation is zero, that is D = 0, then the root of the equation is a real number. Because when D = 0, x = (-b ± √0) / 2a and the square root of 0 is 0, so x = -b/2a. 

The value of the discriminant can predict the nature of the roots. The value of D can be positive, negative, or zero. Here are the differences between positive, zero, and negative discriminants. 
 

The discriminant is a powerful tool in quadratic equations that helps predict the nature, type, and sometimes even the value of roots without fully solving the equation. These tips and tricks make it quick and easy to use.

 

1. Discriminant is D = b² −4ac; identify a, b, c carefully to avoid mistakes.
   
    
2. If D > 0 is two real roots, D = 0 is one real roots, D < 0 is two complex roots.
   
    
3. If D is a perfect square, roots are rational; otherwise, they are irrational.
   
    
4. A negative D means no real roots, saving you from solving the quadratic.
   
    
5. Perfect-square D suggests easy factorization; non-perfect-square D needs the quadratic formula.

In mathematics, the discriminant is a fundamental concept used mainly in quadratic equations. It is also used in the fields of engineering, physics, computer graphics, economics, etc. Here are the applications of discriminant in our real life. 

1. Engineering and Physics: The discriminant helps determine if a quadratic equation describing motion or forces has real solutions, predicting outcomes like projectile landing points.
   
    
2. Finance and Economics: Used to analyze profit and cost functions modeled by quadratics, the discriminant shows whether break-even points exist and how many there are.
   
    
3. Architecture and Design: In structural design, parabolic curves often model arches or bridges; the discriminant helps check if certain dimensions produce real feasible intersections.
   
    
4. Computer Graphics: When calculating ray-object intersections in 3D graphics, the discriminant indicates whether rays intersect surfaces, touch tangentially, or miss entirely.
   
    
5. Robotics and Navigation: Quadratic equations often arise in trajectory planning. The discriminant helps decide whether a robot can reach a specific point or if the path is impossible.