What is Discriminant?
In the case of quadratic equations, the discriminant is commonly employed to determine the nature of the roots. Though determining a discriminant for any polynomial is difficult, we may use formulas to get the discriminant of quadratic and cubic equations.
In arithmetic, a polynomial's discriminant is a function of the polynomial's coefficients. It's useful for figuring out what kind of solutions a polynomial equation has without having to locate them. The name "discriminant" comes from the fact that it distinguishes between the equation's solutions (as equal and unequal; real and nonreal).
It is commonly denoted by Δ or D. The discriminant's value can be any real number (i.e., either positive, negative, or 0).
Quadratic means a variable that is multiplied by itself. The operation essentially includes squaring. A general quadratic equation is –
ax2 + bx + c = 0
With the help of this formula, the roots of a quadratic equation can be found. This root pertains to the value represented by ‘x’.
Formula and Relationship between Roots and Discriminant
Any polynomial's discriminant (Δ or D) is defined in terms of its coefficients. 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 known as the roots of the quadratic equation ax2 + bx + c = 0.
To find them, use the quadratic formula:
X = \[\frac{-b\pm \sqrt{D}}{2a}\]
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 two real roots to the quadratic equation if
D > 0.
This is because the roots of D > 0 are provided by x =
\[\frac{-b\pm \sqrt{\textrm{Positive number}}}{2a}\]
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 =
\[\frac{-b\pm \sqrt{\textrm{Negative number}}}{2a}\]
and so when you take the square root of a negative number, you always get an imaginary number.
If discriminant is equal to zero:
The quadratic equation has two equal real roots if D = 0.
This is because the roots of D = 0 are provided by x = \[\frac{-b\pm \sqrt{0}}{2a}\]
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 zero, it has only one real root.
For example, the given quadratic equation is –
6x2 + 10x – 1 = 0
From the above equation, it can be seen that:
a = 6,
b = 10,
c = −1
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.
Things to Remember While Using Quadratic Formula
It is absolutely necessary that the arrangement of the equation is made in a correct manner, else the solution cannot be obtained.
Ensure that 2a and the square root of the entire (b2 − 4ac) is placed at the denominator.
Keep an eye out for negative b2. Since it cannot be negative, be sure to change it to positive. The square of either positive or negative will always be positive.
Retain the +/−. Watch out for two solutions.
While using a calculator, the number will have to be rounded on a specific number of decimal places.
FAQs on Discriminant
1.Why are the formula of Discriminant important?
The discriminant is a part of the quadratic formula that appears below the square root. A quadratic equation's discriminant is significant since it reveals the number and kind of solutions.
This knowledge is useful since it acts as a double check when utilising any of the four ways to solve quadratic equations (factoring, completing the square, using square roots, and using the quadratic formula).
Also, when we only need to know if a quadratic equation has two real solutions, one real solution, or two complex solutions, it is quite beneficial.
2.Why should you learn how to solve quadratic function problems?
A bow and arrow are launched into the air. How far will it rise? When it lands, how far away will it be? It turns out that a basic understanding of quadratic functions is all you need to answer these and related topics.
If you want to know where an object that is thrown, shot, or launched near the Earth's surface is at all times, quadratic functions can be used to do this. A quadratic function can be used to describe the flight route as long as wind resistance isn't a major factor and the distances aren't too vast.
3.What do you understand about the nature of roots?
The category in which the roots fall merely refers to the nature of the roots. The roots could be made up real, unequal, or even equal. The roots will be fictional if the discriminant is negative.
Calculate the discriminant value of a cubic equation to discover the nature of its roots. The cubic equation has real roots if the discriminant is zero and all the coefficients of the cubic equations are real.
4.What are the three forms of a quadratic equation?
There are three commonly used forms of quadratics:
1. Standard Form:
Y = ax2 + bx + c
The standard form has the advantage of easily recognising a function's end behaviour as well as the values of a, b, and c.
2. Factored Form:
Y = a(x − r1) (x−r2)
We may also use the value of ‘a’ to determine end behaviour in the factored form of a quadratic. Although the degree isn't immediately recognised, we know there are just two components that make the degree two.
3. Vertex Form:
Y = a(x−h)2 + k
The vertex form of a quadratic is what we have here. Remember that the vertex is the point on the parabola where the axis of symmetry intersects. It's also the lowest point of an opening up parabola or the highest point of an opening down parabola.
5.Without calculating, determine how many real solutions the equation 3x2 − 2x + 1 has?
Add 1 to both sides of the quadratic equation to make it equal to 0.
3x2 − 2x + 1= 0
Set a = 3, b = −2, and c = 1, and let’s evaluate the discriminant.
b2 − 4ac
=(−2)2 − 4(3)(1)
= 4 − 12
= −8
The quadratic equation has no real number solutions, just two imaginary ones because the discriminant is negative.