Nature of Roots based on Discriminant

The “Nature of Roots” of a quadratic equation refers to the characteristics of the solutions (also called roots) of the equation. A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$. The nature of the roots is determined by the discriminant, denoted by the letter $D$. The discriminant helps us understand whether the roots are real and distinct, real and equal, or not real (complex). It’s the part under the square root in the quadratic formula.

Formulae

The discriminant, $D$, is calculated as follows:

$$D = b^2 – 4ac$$

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

• If $D > 0$: The equation has two distinct real roots.
• If $D = 0$: The equation has two real and equal roots (a repeated root).
• If $D < 0$: The equation has no real roots. The roots are complex conjugates.

Examples

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

Here, $a = 1$, $b = -5$, and $c = 6$.

Calculate the discriminant: $D = (-5)^2 – 4(1)(6) = 25 – 24 = 1$.

Since $D > 0$, the equation has two distinct real roots. The roots are $x = 2$ and $x = 3$ (which can be found by factoring or using the quadratic formula).

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

Here, $a = 1$, $b = 4$, and $c = 4$.

Calculate the discriminant: $D = (4)^2 – 4(1)(4) = 16 – 16 = 0$.

Since $D = 0$, the equation has two real and equal roots. The root is $x = -2$ (repeated).

Theorem with Proof

Theorem: The nature of the roots of the quadratic equation $ax^2 + bx + c = 0$ is determined by the discriminant $D = b^2 – 4ac$ as follows: If $D > 0$, there are two distinct real roots; if $D = 0$, there are two real and equal roots; if $D < 0$, there are no real roots.

Proof:

The quadratic formula provides the solutions (roots) of a quadratic equation:

$$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$

Observe the term inside the square root: $b^2 – 4ac$. This is the discriminant, $D$.

• If $D > 0$, then $\sqrt{D}$ is a positive real number. Therefore, there are two distinct real solutions, one obtained by adding $\sqrt{D}$ to $-b$ and the other by subtracting $\sqrt{D}$ from $-b$.
• If $D = 0$, then $\sqrt{D} = 0$. Therefore, the quadratic formula simplifies to $x = \frac{-b}{2a}$, which gives a single real solution (or two equal real roots).
• If $D < 0$, then $\sqrt{D}$ is the square root of a negative number. This results in complex numbers, meaning there are no real roots. The solutions are complex conjugates.

Common mistakes by students

• Incorrectly calculating the discriminant: Students often make errors in the arithmetic when calculating $b^2 – 4ac$. Be very careful with signs and order of operations.
• Forgetting the role of ‘a’: Students sometimes forget to include the ‘a’ value (the coefficient of $x^2$) in the calculation of the discriminant.
• Misinterpreting the discriminant: Students sometimes incorrectly associate the value of D with the roots. Remember, D tells us about the nature of the roots (real, equal, not real), not the roots themselves (unless D=0).
• Not recognizing the need to rewrite the equation: Students might attempt to calculate the discriminant before properly rearranging the quadratic equation to the standard form $ax^2 + bx + c = 0$.

Real Life Application

The concept of the nature of roots can be applied in various real-life scenarios, particularly when modeling situations that involve quadratic relationships. For example:

• Projectile Motion: When analyzing the trajectory of a ball thrown in the air, the discriminant can determine whether the ball hits the ground (real roots) or if the mathematical model (perhaps an oversimplification) leads to an unrealistic result (no real roots).
• Optimization Problems: Finding the maximum or minimum value of a quadratic function, such as maximizing profit in a business, uses techniques related to quadratic equations and, implicitly, the discriminant to see when the maximum profit actually occurs.
• Engineering and Design: In civil engineering (e.g., bridge design), or any construction projects that require quadratic curves, the discriminant helps to ensure that certain parameters are achievable in the real world.

Fun Fact

The discriminant can be used to find perfect square trinomials. A perfect square trinomial is a trinomial that can be factored into $(ax + b)^2$ or $(ax – b)^2$. The discriminant of a perfect square trinomial will always be equal to zero! For example, $x^2 + 6x + 9$ factors to $(x + 3)^2$ and its discriminant is $(6)^2 – 4(1)(9) = 36 – 36 = 0$.

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Q.1 For what value of $k$ will the quadratic equation $2x^2 + kx + 3 = 0$ have equal real roots?

Check Solution

Ans: A

For equal roots, the discriminant $D = b^2 – 4ac$ must equal 0. Thus, $k^2 – 4(2)(3) = 0$, which simplifies to $k^2 = 24$. Therefore, $k = \pm \sqrt{24} = \pm 2\sqrt{6}$.

Q.2 The quadratic equation $x^2 – 4x + k = 0$ has distinct real roots. Which of the following is a possible value for $k$?

Check Solution

Ans: C

For distinct real roots, $D > 0$. Therefore, $(-4)^2 – 4(1)(k) > 0$, which simplifies to $16 – 4k > 0$, or $4k < 16$, so $k < 4$. Only 3 satisfies this condition.

Q.3 If the discriminant of a quadratic equation is negative, then the roots are:

Check Solution

Ans: C

A negative discriminant implies that the square root in the quadratic formula is imaginary, meaning the roots are not real.

Q.4 The quadratic equation $x^2 + 6x + 9 = 0$ has roots that are:

Check Solution

Ans: B

The discriminant is $6^2 – 4(1)(9) = 36 – 36 = 0$. A zero discriminant implies equal real roots.

Q.5 For what values of $m$ does the equation $mx^2 + 2x + 3 = 0$ have no real roots?

Check Solution

Ans: A

For no real roots, $D < 0$. Thus, $2^2 - 4(m)(3) < 0$, which simplifies to $4 - 12m < 0$, or $12m > 4$, so $m > \frac{1}{3}$.

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