Discriminant Formula

Discriminant Formula

The Discriminant Formula is used to determine how many solutions a quadratic problem has. In Algebra, the Discriminant Formula is the term that occurs in the quadratic formula under the square root (radical) sign.

A polynomial’s Discriminant Formula is a function of its coefficients and is symbolised by the capital ‘D’ or Delta sign (Δ). Discriminant Formula demonstrates the nature of the roots of any quadratic equation with rational values a, b, and c. A quadratic equation may simply indicate the actual roots or the number of x-intercepts. This formula is used to determine if the quadratic equation’s roots are real or imaginary. Let’s learn more about discriminant formula in this article.

What is a Discriminant?

In mathematics, the discriminant of a polynomial is a function of its coefficients. It helps determine the nature of the solutions of a polynomial equation without actually solving it. The discriminant distinguishes between different types of solutions (such as equal and unequal, real and non-real), which is why it is called the “discriminant.” It is typically represented by Δ or D. The discriminant’s value can be any real number, meaning it can be positive, negative, or zero.

Formula for Discriminant

Discriminant formulas are essential for understanding and analyzing polynomial equations, especially quadratic and cubic equations. The discriminant offers crucial information about the nature of the roots of the equation. Here is a discriminant formula for a quadratic and cubic equation:

For a quadratic equation in the form 

ax²+bx+c=0, the discriminant (Δ) is determined using the following formula:

Δ=b² −4ac

For a cubic equation in the form 

ax³ + bx² + cx + d = 0, the discriminant (Δ) is determined using the following formula:

Δ = b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd

How to Find Discriminant?

To find the discriminant formula of any equation, follow these steps:

• Identify the coefficients
• Use the discriminant formula
• Substitute the coefficients
• Calculate the result

Discriminant Formulas of a Quadratic Equation

The discriminant of a quadratic equation ax² +bx+c=0 is expressed in terms of its coefficients 

a, b, and c. The formula for the discriminant (denoted as Δ or D) is:

Δ or D=b² −4ac

If Δ>0: The equation has two distinct real roots.This is because, when D > 0, the roots are given by x = (−b±√Positive number)/2a, and the square root of a positive number always results in a real number.

If Δ=0: The equation has a single real root, also known as a repeated or double root.

If Δ<0: The equation has two complex roots (conjugate pairs), which are not real.This is because, when D < 0, the roots are given by x = −b±√ Negative number /2a and the square root of a negative number always leads to an imaginary number.

When the discriminant (Δ) of a quadratic equation is negative, it indicates that the equation has no real roots. Instead, it has two distinct, complex roots. These complex roots are generally expressed in the form a±bi, where “a” and “b” are real numbers, and “i” represents the imaginary unit (the square root of -1).

Discriminant Formulas of a Cubic Equation

The discriminant formula for a cubic equation ax³ + bx² + cx + d = 0is given by 

Δ (or D): b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd

A cubic equation can have up to three roots because its degree is 3. The nature of these roots depends on the value of Δ:

If Δ>0, all three roots are real and distinct.

If Δ=0, all three roots are real, with at least two of them being equal.

If Δ<0, two of the roots are complex numbers (conjugate pairs), and the third root is real.

Why is Discriminant Formula Important?

The discriminant formula can be used to determine the number of roots in a quadratic equation. The discriminant can be positive, negative, or zero, and the value of the discriminant reveals the nature of the roots as follows:

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• If the discriminant is zero, the quadratic equation has one real solution, which is a repeated (or double) root.
• If the discriminant is negative, the quadratic equation has no real solutions, but instead has two complex (imaginary) solutions.

Discriminant Formula Applications

The discriminant is a key mathematical tool, particularly in the context of quadratic equations. It offers crucial insights into the nature of the solutions (roots) of a quadratic equation. Here are several applications:

Nature of Roots: The discriminant helps determine the type of solution to a quadratic equation.

Quadratic Equations: The discriminant is extensively used in solving and analyzing quadratic equations of the form ax² + bx + c = 0

Geometry: In geometry, the discriminant can be used to analyze conic sections.

Optimization Problems: In optimization problems, the discriminant is useful for finding the maximum or minimum values of a quadratic function.

Computer Graphics: In computer graphics and 3D modeling, quadratic equations are used to represent curves and surfaces.

Solved Examples on Discriminant Formula

Example 1: Calculate the discriminant for the equation: √3x² + 10x − 8√3 = 0. 

Solution:

Given the quadratic equation: 

√3x² + 10x − 8√3 = 0

By comparing this with the standard quadratic form ax² + bx + c = 0, we identify a = √3, b = 10, and c = -8√3.

Using the discriminant formula:

Δ=b² − 4ac

Substituting the values of a, b, and c into the formula:

Δ(10)² – 4(√3)(-8√3)

=100−4⋅(−24)=100+96

Therefore, the discriminant is:

Δ=100+96=196

So, the discriminant of the equation √3x² + 10x − 8√3 =0 is 196.

Example 2: Determine the discriminant of the equation x2 −2x+3=0. Also, identify the number of solutions for this equation.

Solution:

Given the quadratic equation: 

x² −2x + 3 = 0

In this equation:

a = 1, b= −2, c = 3

To find the discriminant, use the formula:

Δ=b² −4ac

Substitute the values of a, b, and c:

Δ=(−2)2 −4(1)(3)

Δ=4−12

Thus, the discriminant is:Δ=−8

Since the discriminant (Δ) is negative, the quadratic equation has no real solutions. Instead, it has two distinct, complex roots.

Example 3: What is the discriminant of quadratic equation 9z² − 6b²z − (a⁴ − b⁴) = 0

Solution:

To find the discriminant, we first compare the given quadratic equation to the standard form 

ax²+bx+c=0. This gives us the coefficients:

a = 9, b = -6b², and c = – (a⁴ − b⁴). Its discriminant is,

Δ = b² – 4ac

= (-6b²)² – 4 (9) [-(a⁴ − b⁴)]

= 36b⁴ + 36a⁴ – 36b⁴

= 36a⁴