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Find the Discriminant and Type of Solutions Calculator

Discriminant

The discriminant is widely used in the case of quadratic equations and is used to find the nature of the roots. Though finding a discriminant for any polynomial is not so easy, there are formulas to find the discriminant of quadratic and cubic equations that make our work easier.

Let us learn more about the discriminant along with its formulas and let us also understand the relation between the discriminant and the nature of the roots.

1. What is Discriminant in Math?
2. Discriminant Formula
3. How to Find Discriminant?
4. Discriminant and Nature of the Roots
5. FAQs on Discriminant

What is Discriminant in Math?

Discriminant of a polynomial in math is a function of the coefficients of the polynomial. It is helpful in determining what type of solutions a polynomial equation has without actually finding them. i.e., it discriminates the solutions of the equation (as equal and unequal; real and nonreal) and hence the name "discriminant". It is usually denoted by Δ or D. The value of the discriminant can be any real number (i.e., either positive, negative, or 0).

Discriminant Formula

The discriminant (Δ or D) of any polynomial is in terms of its coefficients. Here are the discriminant formulas for a cubic equation and quadratic equation.

Discriminant formula is shown for both quadratic equation and cubic equation. The discriminant formulas are in terms of coefficients of polynomial.

Let us see how to use these formulas to find the discriminant.

How to Find Discriminant?

To find the discriminant of a cubic equation or a quadratic equation, we just have to compare the given equation with its standard form and determine the coefficients first. Then we substitute the coefficients in the relevant formula to find the discriminant.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation ax2 + bx + c = 0 is in terms of its coefficients a, b, and c. i.e.,

  • Δ OR D = b2 − 4ac

Do you recall using b2 − 4ac earlier? Yes, it is a part of the quadratic formula: x = \(\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\). Here, the expression that is inside the square root of the quadratic formula is called the discriminant of the quadratic equation. The quadratic formula in terms of the discriminant is: x = \(\dfrac{-b \pm \sqrt{D}}{2 a}\).

Example: Find the discriminant of the quadratic equation 2x2 - 3x + 8 = 0.

Comparing the equation with ax2 + bx + c = 0, we get a = 2, b = -3, and c = 8. So the discriminant is,
Δ OR D = b2 − 4ac = (-3)2 - 4(2)(8) = 9 - 64 = -55.

Discriminant of Cubic Equation

The discriminant of a cubic equation ax3 + bx2 + cx + d = 0 is in terms of a, b, c, and d. i.e.,

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

Example: Find the discriminant of the cubic equation x3 - 3x + 2 = 0.

Comparing the equation with ax3 + bx2 + cx + d = 0, we have a = 1, b = 0, c = -3, and d = 2. So its discriminant is,

Δ or D = b2c2 − 4ac3 − 4b3d − 27a2d2 + 18abcd
= (0)2(-3)2 − 4(1)(-3)3 − 4(0)3(2) − 27(1)2(2)2 + 18(1)(0)(-3)(2)
= 0 + 108 - 0 - 108 + 0
= 0

Discriminant and Nature of the Roots

The roots of a quadratic equation ax2 + bx + c = 0 are the values of x that satisfy the equation. They can be found using the quadratic formula: x = \(\dfrac{-b \pm \sqrt{D}}{2 a}\). Though we cannot find the roots by just using the discriminant, we can determine the nature of the roots as follows.

If Discriminant is Positive

If D > 0, the quadratic equation has two different real roots. This is because, when D > 0, the roots are given by x = \(\dfrac{-b \pm \sqrt{\text { Positive number }}}{2 a}\) and the square root of a positive number always results in a real number. So when the discriminant of a quadratic equation is greater than 0, it has two roots which are distinct and real numbers.

If Discriminant is Negative

If D < 0, the quadratic equation has two different complex roots. This is because, when D < 0, the roots are given by x = \(\dfrac{-b \pm \sqrt{\text { Negative number }}}{2 a}\) and the square root of a negative number leads to an imaginary number always. For example \(\sqrt{-4}\) = 2i. So when the discriminant of a quadratic equation is less than 0, it has two roots which are distinct and complex numbers (non-real).

If Discriminant is Equal to Zero

If D = 0, the quadratic equation has two equal real roots. In other words, when D = 0, the quadratic equation has only one real root. This is because, when D = 0, the roots are given by x = \(\dfrac{-b \pm \sqrt{\text { 0 }}}{2 a}\) and the square root of a 0 is 0. Then the equation turns into x = -b/2a which is only one number. So when the discriminant of a quadratic equation is zero, it has only one real root.

A root is nothing but the x-coordinate of the x-intercept of the quadratic function. The graph of a quadratic function in each of these 3 cases can be as follows.

The relation between discriminant and the roots of a quadratic equation is shown by using a graph when D is greater than 0, less than 0, and equal to 0.

Important Notes on Discriminant:

  • The discriminant of a quadratic equation ax2 + bx + c = 0 is Δ OR D = b2 − 4ac.

  • A quadratic equation with discriminant D has:
    (i) two unequal real roots when D > 0
    (ii) only one real root when D = 0
    (iii) no real roots or two complex roots when D < 0

Related Topics:

  • Solving Quadratic Equations
  • Discriminant Calculator
  • Factoring Quadratics
  • Quadratic Expressions
  • Quadratic Function

Discriminant Examples

  1. Example 1: Find the discriminant of the following equation: √3x2 + 10x − 8√3 = 0.

    Solution:

    The given quadratic equation is √3x2 + 10x − 8√3 = 0. Comparing this with ax2 + bx + c = 0, we get a = √3, b = 10, and c = -8√3.

    The quadratic discriminant formula is:

    D = b2 - 4ac
    = (10)2 - 4(√3)(-8√3)
    = 100 + 96
    = 196

    Answer: The discriminant = 196.

  2. Example 2: Determine whether each of the following quadratic equation has two real roots, one real root, or no real roots. (a) 3x2 − 5x − 7 = 0 (b) 2x2 + 3x + 3=0.

    Solution:

    (a) By comparing the given equation with ax2 + bx + c = 0, we get a = 3, b = -5, and c = -7. Its discriminant is,

    D = b2 - 4ac
    = (-5)2 - 4(3)(-7)
    = 25 + 84
    = 109

    So we have got a positive discriminant and hence the given equation has two real roots.

    (b) By comparing the given equation with ax2 + bx + c = 0, we get a = 2, b = 3, and c = 3. Its discriminant is,

    D = b2 - 4ac
    = (3)2 - 4(2)(3)
    = 9 - 24
    = -15

    So we have got a negative discriminant and hence the given equation has no real roots.

    Answer: (a) Two real roots (b) No real roots.

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

    Solution:

    By comparing the given equation with ax2 + bx + c = 0, we get a = 9, b = -6b2, and c = - (a4 − b4). Its discriminant is,

    D = b2 - 4ac
    = (-6b2)2 - 4 (9) [-(a4 − b4)]
    = 36b4 + 36a4 - 36b4
    = 36a4

    Answer: The discriminant of the given quadratic equation is 36a4.

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Practice Questions on Discriminant

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FAQs on Discriminant

What is Discriminant Meaning?

The discriminant in math is defined for polynomials and it is a function of coefficients of polynomials. It tells the nature of roots or in other words, it discriminates the roots. For example, the discriminant of a quadratic equation is used to find:

  • How many roots it has?
  • Whether the roots are real or non-real?

What is Discriminant Formula?

There re different discriminant formulas for different polynomials:

  • The discriminant of a quadratic equation ax2 + bx + c = 0 is Δ OR D = b2 − 4ac.

  • The discriminant of a cubic equation ax3 + bx2 + cx + d = 0 is Δ or D = b2c2 − 4ac3 − 4b3d − 27a2d2 + 18abcd.

How to Calculate the Discriminant of a Quadratic Equation?

To calculate the discriminant of a quadratic equation:

  • Identify a, b, and c by comparing the given equation with ax2 + bx + c = 0.
  • Substitute the values in the discriminant formula D = b2 − 4ac.

What if Discriminant = 0?

If the discriminant of a quadratic equation ax2 + bx + c = 0 is 0 (i.e., if b2 - 4ac = 0), then the quadratic formula becomes x = -b/2a and hence the quadratic equation has only one real root.

What Does Positive Discriminant Tell Us?

If the discriminant of a quadratic equation ax2 + bx + c = 0 is positive (i.e., if b2 - 4ac > 0), then the quadratic formula becomes x = (-b ± √(positive number) ) / 2a and hence the quadratic equation has only two real and distinct roots.

What Does Negative Discriminant Tell Us?

If the discriminant of a quadratic equation ax2 + bx + c = 0 is negative (i.e., if b2 - 4ac < 0), then the quadratic formula becomes x = (-b ± √(negative number) ) / 2a and hence the quadratic equation has only two complex and distinct roots.

What is the Formula for Discriminant of Cubic Equation?

A cubic equation is of the form ax3 + bx2 + cx + d = 0 and its discriminant is in terms of its coefficients which is given by the formula D = b2c2 − 4ac3 − 4b3d − 27a2d2 + 18abcd.

Find the Discriminant and Type of Solutions Calculator

Source: https://www.cuemath.com/algebra/discriminant/