Math Calculators ▶ Discriminant Calculator
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An online discriminant calculator helps to find the discriminant of the quadratic polynomial as well as higher degree polynomials. You can try this discriminant finder to find out the exact nature of roots and the number of root of the given equation.
Well, give a thorough read to know about each and everything related to discriminant calculations. So, let’s make a start from the basic definition.
In maths, a discriminant is a function of coefficients of the polynomial equation that displays the nature of the roots of a given equation. It is represented by a \(Δ\) sign (read as delta). If you have a concern with the term “what does the discriminant tell you”, then keep reading.
And, make a click on this online quadratic formula calculator that works best to solve the quadratic equation by using a quadratic formula or complete the square method.
The discriminant of the quadratic equation determines the roots’ nature.
The discriminant of the cubic equation determines the roots’ nature.
The discriminant of the quartic equation determines the roots’ nature.
The discriminant of the quadratic equation determines how many roots are there in an equation.
The discriminant of the cubic equation determines how many roots are there in an equation.
The discriminant of the quartic equation determines how many roots are there in an equation.
The online discriminant calculator shows the nature of roots of the quartic equation and if you want to determine the nature of roots for the cubic and quadratic equations, then this online tool is handy.
The discriminant of an equation determine the shape of the parabola in a graph,
The standard formula for the following standard polynomial equation is:
$$ p(x) = a_nx^n + . . . + a_1x + a_0 $$
the equation has exactly \(n\) roots \(x_1, . . . , x_n\) (remember that these roots not necessarily all unique! Now, here we figure out the discriminant of \(p\) as:
$$ D(p) = a_n \text{ }^{2n-2} \prod (x_i – x_j)^2 $$
where;
the product \(\prod\) is taken over all \(i < j\)
The standard discriminant form for the quadratic, cubic, and quartic equations is as follow,
The standard discriminant formula for the quadratic equation \(ax^2 + bx + c = 0\) is,
$$ Δ = b^2-4ac $$
Where,
The standard discriminant form for the cubic equation \(ax^3 + bx^2 + cx + d = 0\) is,
\(Δ=b^2c^2 – 4ac^3-4b^3d-27a^2d^2+18abcd\)
Where,
The standard discriminant form for the quartic equation \(ax^4 + bx^3 + cx^2 + dx + e = 0\) is,
\(Δ = 256a^3e^3 – 192a^2bde^2 – 128a^2c^2e^2 + 144a^2cd^2e – 27a^2d^4 + 144ab^2ce^2 – 6ab^2d^2e\)\( – 80abc^2de +18abcd^3 + 16ac^4e – 4ac^3d^2 – 27b^4e^2 +18b^3cde – 4b^3d^3 – 4b^2c^3e + b^2c^2d^2\)
Where,
As we know the discriminant of a quadratic equation has only two terms, but as the degree of polynomial increases, the discriminant becomes more complicated.
Our online discriminant calculator finds out the determinant of quadratic, cubic, and quartic equations.
An equation having the highest exponent of a variable (usually x) is 2.It is also called the second-degree equation. As from its name, “quadratic” comes from the word ‘quad’ means square because the variable gets squared in this equation.
Standard form of a quadratic equation is,
$$ ax^2 + bx + c = 0 $$
Where,
\(a\), \(b\) and \(c\) are known numbers while \(x\) is an unknown variable \((a≠0)\).
A polynomial equation in which the highest exponent of a variable (usually x) is three.
Standard form of a cubic equation is,
$$ ax^3 + bx^2 + cx + d = 0 $$
Where,
\(a\), \(b\), \(c\) and \(d\) are known numbers while \(x\) is an unknown variable \((a≠0)\).
A polynomial equation in which the highest exponent of a variable(usually x)is four.
The derivative of the quartic equation is a cubic function.
Standard form of a quartic equation is,
$$ ax^4 + bx^3 + cx^2 + dx + e = 0 $$
Where,
\(a\), \(b\), \(c\), \(d\) and \(e\) are known numbers while \(x\) is an unknown variable \((a≠0)\).
The discriminant calculator shows you the step-by-step calculations for the given equation problems. It doesn’t matter whether you want to calculate quadratic equation and higher degree polynomials equation, this calculator does all for you! To finding the discriminant of the equations, just follow these inputs:
Inputs:
Outputs:
The discriminant calculator will find:
Let’s have an example of each type of equation and have step by step calculations for each.
The formula for the discriminant of quadratic equation is,
$$ Δ = b^2-4ac $$
For example:
If we have an equation, \(3x^2+2x-9=0\), then find the discriminant?
Solution:
Here, \(a = 3\)
\(b = 2\)
\(c = -9\)
Putting the values in the given formula,
\(Δ = (2)^2-4(3)(-9)\)
\(Δ = 4+108\)
\(Δ = 112\)
The formula for the discriminant of cubic equation is,
$$ Δ= b^2c^2 – 4ac^3-4b^3d-27a^2d^2+18abcd $$
For example:
Calculate the discriminant of the following equation?
$$ 5x^3 + 2x^2 + 8x + 6 = 0 $$
Solution:
Here, \(a = 5\)
\(b = 2\)
\(c = 8\)
\(d = 6\)
Putting the values in the given formula,
\(Δ=(2)^2(8)^2 – 4(5)(8)^3-4(2)^3(6)-27(5)^2(6)^2+18(5)(2)(8)(6)\)
\(Δ=(4)(64) – 4(5)(512)-4(8)(6)-27(25)(36)+18(480)\)
\(Δ=(4)(64) – 4(2560)-4(48)-27(900)+18(480)\)
\(Δ=256 – 10240-192-24300+8640\)
\(Δ=8896 – 10240-192-24300\)
\(Δ=8896 – 10240-24492\)
\(Δ=8896 – 34732\)
\(Δ= -25836\)
The formula for the discriminant of quartic equation is,
\(Δ=256a^3e^3 – 192a^2bde^2-128a^2c^2e^2 + 144a^2cd^2e -27a^2d^4 + 144ab^2ce^2 – 6ab^2d^2e – 80abc^2de\) \(+18abcd^3 + 16ac^4e – 4ac^3d^2 – 27b^4e^2 +18b^3cde – 4b^3d^3 – 4b^2c^3e + b^2c^2²d^2\)
For example:
Calculate the discriminant of the following equation?
\(2x^4+x^3+3x^2+2x+7=0\)
Solution:
Here, \(a = 2\)
\(b = 1\)
\(c = 3\)
\(d = 2\)
\(e = 7\)
Putting the values in the given formula,
\(= 256 \times (2)^3 \times (7)^3 – (192) \times (2)^2 \times 1 \times 2 \times (7)^2 – (128) \times (2)^2 \times (3)^2 \times (7)^2 + (144) \times (2)^2 \times 3 \times (2)^2 \times 7 – (27) \times (2)^2 \times (2)^4\)
\( + (144) \times 2 \times (1)^2 \times 3 \times (7)^2 – (6) \times 2 \times (1)^2 \times (2)^2 \times 7 – (80) \times 2 \times 1 \times (3)^2 \times 2 \times 7 + (18) \times 2 \times 1 \times 3 \times (2)^3\)
\( + (16) \times 2 \times (3)^4 \times 7 – (4) \times 2 \times (3)^3 \times (2)^2 – (27) \times (1)^4 \times (7)^2+ (18) \times (1)^3 \times 3 \times 2 \times 7 – (4) \times (1)^3 \times (2)^3 \)
\(- (4) \times (1)^2 \times (3)^3 \times 7 + (1)^2 \times (3)^2 \times (2)^2\)
\(= (256 \times 8 \times 343) – (192 \times 4 \times 1 \times 2 \times 49) – (128 \times 4 \times 9 \times 49) + (144 \times 4 \times 3 \times 4 \times 7)\)
\(- (27 \times 4 \times 16) + (144 \times 2 \times 1 \times 3 \times 49) – (6 \times 2 \times 1 \times 4 \times 7 ) – (80 \times 2 \times 1 \times 9 \times 2 \times 7)\)
\(+ (18 \times 2 \times 1 \times 3 \times 8) + (16 \times 2 \times 81 \times 7) -(4 \times 2 \times 27 \times 4 )- (27 \times 1 \times 49)+ (18 \times 1 \times 3 \times 2 \times 7)\)
\(- (4 \times 1 \times 8) – (4 \times 1 \times 27 \times 7) + (1 \times 9 \times 4)\)
\(= 702464 – 75264 – 225792 + 48384 – 1728 + 42336 – 336 – 20160 + 864 – 18144 – 864 + 1323 + 756 – 32 – 756 + 36\)
\(= 453087\)
Yes, the value can be negative or positive. A negative value for the discriminant indicates the roots are complex while a positive value returns the real roots.
If the discriminant is zero, then the number of roots is two (2).
If the discriminant of the quadratic equation, the part under the square root is negative, then the roots are imaginary.
To analyze discriminant in excel, just follow the given steps,
If you want to know about the nature of the roots of an equation, then you should find out the discriminant of that equation. So, try an online discriminant calculator that allows you to find out the nature root of the equation with complete calculation.
From the authorized source of Wikipedia: Discriminant definition, Expression in terms of the roots and all you need to know about that!
From the source of khanacademy (CCSS.Math: HSA.REI.B.4, HSA.REI.B.4b): A detailed review on Discriminant
The brilliant source provided with: Finding the Discriminant of a Quadratic (Explanation), Repeated Roots and Range of Solutions
From the source of studypug: Nature of roots of quadratic equations: