Discriminant of a Quadratic Equation

The discriminant is a crucial concept in algebra, particularly when dealing with quadratic equations. It provides valuable information about the nature of the roots (solutions) of a quadratic equation without actually solving the equation itself. The discriminant is calculated using the coefficients of the quadratic equation, which is in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \ne 0$. The discriminant helps determine whether the roots are real, complex, and whether there are two distinct roots, one repeated root, or no real roots.

Formulae

The discriminant, denoted by the letter $D$, is defined by the following formula:

$$D = b^2 – 4ac$$

Based on the value of $D$, we can determine the nature of the roots:

• If $D > 0$: The quadratic equation has two distinct real roots.
• If $D = 0$: The quadratic equation has one real root (a repeated root).
• If $D < 0$: The quadratic equation has no real roots; the roots are complex (conjugate pairs).

Examples

Here are some examples illustrating the use of the discriminant:

Example-1: Consider the quadratic equation $x^2 + 5x + 6 = 0$.

Here, $a = 1$, $b = 5$, and $c = 6$. The discriminant is:

$$D = (5)^2 – 4(1)(6) = 25 – 24 = 1$$

Since $D > 0$, the equation has two distinct real roots. (In this case, the roots are -2 and -3.)

Example-2: Consider the quadratic equation $x^2 + 4x + 4 = 0$.

Here, $a = 1$, $b = 4$, and $c = 4$. The discriminant is:

$$D = (4)^2 – 4(1)(4) = 16 – 16 = 0$$

Since $D = 0$, the equation has one real root (a repeated root). (In this case, the root is -2.)

Common mistakes by students

• Incorrectly identifying a, b, and c: Students may misidentify the coefficients $a$, $b$, and $c$ from the quadratic equation. Always ensure the equation is in standard form before extracting the coefficients.
• Arithmetic errors: Careless mistakes when calculating $b^2$ or multiplying $4ac$. Double-check your calculations.
• Misinterpreting the discriminant: Confusing the relationship between the value of the discriminant and the number and type of roots. Remember: $D>0$ means two distinct real roots; $D=0$ means one repeated real root; $D<0$ means two complex conjugate roots.
• Not understanding the standard form: Assuming the quadratic equation is always provided in the correct form. Rearranging terms to $ax^2 + bx + c = 0$ is critical for correctly identifying a, b and c.

Real Life Application

The discriminant, and quadratic equations in general, have numerous real-world applications. Here are some examples:

• Physics: Projectile motion. The discriminant helps determine if a projectile (e.g., a ball thrown in the air) will reach a certain height or a certain distance. The maximum height or range can also be obtained with the application of discriminant.
• Engineering: Design of structures like bridges and arches. Quadratic equations are used to model the shape and stability of these structures, and the discriminant can be used to analyze their behavior under stress.
• Business and Economics: Modeling cost and revenue functions, which are often represented by quadratic equations. The discriminant helps determine profit/loss break-even points and other important economic indicators.

Fun Fact

The discriminant is closely related to the quadratic formula. In fact, the discriminant is the part of the quadratic formula that lies under the square root sign. Therefore, the value of the discriminant directly determines the nature of the roots found by the quadratic formula.

Recommended YouTube Videos for Deeper Understanding

Q.1 If the discriminant of a quadratic equation is positive, what can be said about the roots?

Check Solution

Ans: C

A positive discriminant indicates that the square root in the quadratic formula will result in two distinct real numbers.

Q.2 What is the discriminant of the quadratic equation $2x^2 + 5x – 3 = 0$?

Check Solution

Ans: A

$D = b^2 – 4ac = 5^2 – 4(2)(-3) = 25 + 24 = 49$

Q.3 For what value of $k$ does the quadratic equation $x^2 + kx + 9 = 0$ have exactly one real root?

Check Solution

Ans: D

For one real root, $D = 0$. Thus, $k^2 – 4(1)(9) = 0$, which simplifies to $k^2 = 36$. Hence, $k = \pm 6$.

Q.4 If the discriminant of a quadratic equation is zero, what type of roots does it have?

Check Solution

Ans: C

A discriminant of zero means the quadratic formula results in only one root, repeated.

Q.5 Determine the nature of the roots of the quadratic equation $x^2 – 4x + 5 = 0$.

Check Solution

Ans: C

$D = b^2 – 4ac = (-4)^2 – 4(1)(5) = 16 – 20 = -4$. Since the discriminant is negative, the roots are complex.

Next Topic: Nature of Roots based on Discriminant