The discriminant formulas areused to find thediscriminant of a polynomial equation. Especially, the discriminant of a quadratic equation isused to determine the number and the nature of the roots. Thediscriminant of a polynomial is a function that is made up of the coefficients of the polynomial. Let us learn thediscriminant formulas along with a few solved examples.
What Is Discriminant Formulas?
Thediscriminant formulas give us an overview of the nature of the roots. Thediscriminant of a quadratic equation is derived from the quadratic formula. Thediscriminant is denoted by D orΔ. Thediscriminant formulas for a quadratic equationand cubic equation are:
Discriminant Formula of a Quadratic Equation
The discriminant formula of a quadratic equationax2 + bx + c = 0 is,Δ (or) D = b2- 4ac. We know that a quadratic equation has a maximum of 2 roots as its degree is 2.We know that the quadratic formula is used to find the roots of a quadratic equation ax2 + bx + c = 0. According to the quadratic formula, the roots can be found using x = [-b±√ (b2- 4ac) ] / [2a]. Here,b2- 4ac is the discriminant D and it is inside the square root. Thus, the quadratic formula becomesx = [-b±√D] / [2a]. Here D can be either > 0, = 0, (or) < 0. Let us determine the nature of the roots in each of these cases.
- If D> 0, then the quadratic formula becomesx = [-b±√(positive number)] / [2a] and hence in this case the quadratic equation has two distinct real roots.
- If D= 0, then the quadratic formula becomesx = [-b] / [2a] and hence in this case the quadratic equation has only one real root.
- If D< 0, then the quadratic formula becomesx = [-b±√(negative number)] / [2a] and hence in this case the quadratic equation has two distinct complex roots (this is because the square root of a negative number results in an imaginary number. For example,√(-4) = 2i).
Discriminant Formula of a Cubic Equation
The discriminant formula of a cubic equation ax3+ bx2+ cx + d = 0 is,Δ (or) D =b2c2− 4ac3− 4b3d − 27a2d2+ 18abcd. We know that a cubic equation has a maximum of 3roots as its degree is 3. Here,
- If D > 0, all the three rootsare real and distinct.
- If D = 0, then all the three roots are real where at least two of them are equal to each other.
- If D < 0, then two of its roots are complex numbers and the third root is real.
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We can see the applications of the discriminant formulas in the following section.
Examples UsingDiscriminant Formulas
Example 1:Determine the discriminant of the quadratic equation 5x2+ 3x + 2 = 0. Also, determine the nature of its roots.
Solution:
The given quadratic equation is5x2+ 3x + 2 = 0.
Comparing this withax2 + bx + c = 0, we get a = 5, b = 3, and c = 2.
Using discriminant formula,
D = b2-4ac
= 32- 4(5)(2)
= 9 - 40
= -31
Answer: The discriminant is -31. This is a negative number and hence the given quadratic equationhas two complex roots.
Example 2:Determine the discriminant of the quadratic equation 2x2+ 8x + 8 = 0. Also, determine the nature of its roots.
Solution:
The given quadratic equation is2x2+ 8x + 8 = 0.
Comparing this withax2 + bx + c = 0, we get a = 2, b = 8, and c = 8.
Using the discriminant formula,
D = b2- 4ac
= 82- 4(2)(8)
= 64 - 64
= 0
Answer:The discriminant is 0 andhence the given quadratic equationhas two complex roots.
Example 3:Determine the nature of the roots of the cubic equation x3- 4x2+ 6x - 4 = 0.
Solution:
The given cubic equation isx3- 4x2+ 6x - 4 = 0.
Comparing this withax3+ bx2+ cx + d = 0, we get a = 1, b = -4, c = 6, and d = -4.
Using the discriminant formula,
D =b2c2− 4ac3− 4b3d − 27a2d2+ 18abcd
=(-4)2(6)2− 4(1)(6)3− 4(-4)3(-4) − 27(1)2(-4)2+ 18(1)(-4)(6)(-4)
= -16
Answer:Since the discriminant is a negative number, the given cubic equation has two complex roots and one real root.
FAQs onDiscriminant Formulas
What AreDiscriminant Formulas?
The discriminant of a polynomial equation is a function which is in terms of its coefficients. Thediscriminant of an equation is used to determine the nature of its roots. Thediscriminant formulas are as follows:
- The discriminant formula of a quadratic equationax2 + bx + c = 0 is,Δ (or) D = b2- 4ac.
- The discriminant formula of a cubic equation ax3+ bx2+ cx + d = 0 is,Δ (or) D =b2c2− 4ac3− 4b3d − 27a2d2+ 18abcd.
How To Derive theDiscriminant Formula of a Quadratic Equation?
Let us derive thediscriminant formula of a quadratic equationax2 + bx + c = 0. By quadratic formula, the solutions of this equation are found usingx = [-b±√ (b2- 4ac) ] / [2a]. Hereb2- 4ac is inside the square root and hence we can determine the nature of the roots by using the properties of the square root (such as the square root of a positive number is a real number, the square root of a negative number is an imaginary number, and the square root of 0 is 0).Thus, the discriminant of the quadratic equation isb2- 4ac.
What Are the Applications of theDiscriminant Formula?
The discriminant formula is used to determine the nature of the roots of a quadratic equation. The discriminant of a quadratic equation ax2 + bx + c = 0 is D =b2- 4ac.
- If D > 0, then the equation has two real distinct roots.
- If D = 0, then the equation has only one real root.
- If D < 0, then the equation has two distinct complex roots.
What Is theDiscriminant Formula of a Cubic equation?
The discriminant formula of a cubic equation ax3+ bx2+ cx + d = 0 is denoted byΔ (or) D and is found using the formulaΔ (or) D =b2c2− 4ac3− 4b3d − 27a2d2+ 18abcd.