Table of Contents
What is Discriminant?
Discriminant – Explanation: A discriminant is a mathematical expression that is used to determine the nature of a particular type of mathematical equation. In most cases, the discriminant is used to determine whether a particular equation has two or three solutions. For equations that have three solutions, the discriminant will be zero. For equations with two solutions, the discriminant will be a positive number.
Formula and Relationship between Roots and Discriminant
A discriminant is a calculation that is used to determine the nature of the roots of a quadratic equation. The discriminant is the result of taking the square root of the coefficient of the x squared term in the equation. The discriminant is then used to determine if the roots of the equation are real or complex. If the discriminant is positive, then the roots are real. If the discriminant is negative, then the roots are complex.
Things to Remember While Using Quadratic Formula
The Quadratic Formula is a mathematical equation that is used to solve quadratic equations. A quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and also x is the variable. The Quadratic Formula can be used to solve any quadratic equation, regardless of the value of a, b, and c.
The Quadratic Formula is as follows:
x = (-b +- sqrt(b^2 – 4ac))/(2a)
To use the Quadratic Formula, you need to first isolate the variable, x, on one side of the equation. To do this, you need to subtract bx from both sides of the equation. This will leave you with ax^2 on one side and also c on the other.
Next, take the square root of both sides of the equation. This will leave you with b^2 on one side and 4ac on the other.
Finally, divide both sides of the equation by 2a. This will leave you with the value of x.
Formula and Relationship between Roots and Discriminant
Any polynomial’s discriminant (Δ or D) is defined in terms of its coefficients. Therefore the discriminant formulas for a cubic equation and a quadratic equation are:
Discriminant formula of a quadratic equation:
ax2 bx + c = 0 is
Δ or D = b2 − 4ac
Discriminant formula of a cubic equation:
ax + bx³ + cx² + d = 0 is
Δ or D = b2c2 − 4ac3 − 4b3d −27a2d2 + 18abcd
Relationship between Roots and Discriminant
The values of x that satisfy the equation are also known as the roots of the quadratic equation ax2 + bx + c = 0.
To find them, use the quadratic formula:
X =
−b±D−−√2a
Although we cannot discover the roots using the discriminant alone, we can determine the nature of the roots in the following way.
If discriminant is positive:
There are also two real roots to the quadratic equation if
D > 0.
Therefore this is because the roots of D > 0 also are provided by x =
−b±Positive number−−−−−−−−−−−−−√2a
And a real number is always the square root of a positive number.
When the discriminant of a quadratic equation exceeds 0, it has two separate and real-number roots.
If discriminant is negative:
The quadratic equation has two different complex roots if
D < 0.
This is because the roots of D < 0 are provided by x =
−b±Negative number−−−−−−−−−−−−−−√2a
and so when you take the square root of a negative number, you always get an imaginary number.
If discriminant is equal to zero:
Therefore the quadratic equation has two equal real roots if D = 0.
This is because the roots of D = 0 are provided by x =
−b±0–√2a
and 0 would be the square root. The equation thus becomes x = −b/2a, which is a single number. When a quadratic equation’s discriminant is 0, it has only one real root.
For example, the given quadratic equation is –
6×2 + 10x – 1 = 0
From the above equation, it can be seen that:
a = 6,
b = 10,
also c = −1
However Applying the numbers in discriminant –
b2 − 4ac
= 102 – 4 (6) (−1) =100 + 24
= 124
Given that, the discriminant amounts to be a positive number, there are two solutions to the quadratic equation.
