NCERT Class X Chapter 4: Quadratic Equation Example 9

NCERT Class X Chapter 4: Quadratic Equation Example 9

Question:

Find the discriminant of the equation 3x2 – 2x + (1/3) = 0 and hence find the nature of its roots. Find them, if they are real..

Given:

The quadratic equation: 3x2 – 2x + 1 3 = 0

To Find:

1. The discriminant of the equation.
2. The nature of its roots.
3. The roots, if they are real.

Formula:

WKT, for a quadratic equation of the form ax2 + bx + c = 0, the discriminant (Δ) is given by:
Δ = b2 - 4ac

WKT, the nature of roots is determined by the value of the discriminant (Δ):
• If Δ > 0, there are two distinct real roots.
• If Δ = 0, there are two equal real roots.
• If Δ < 0, there are no real roots (or two distinct complex roots).

WKT, if the roots are real, they can be found using the quadratic formula:
x = -b ± √Δ 2a

Solution:

The given quadratic equation is 3x2 – 2x + 1 3 = 0.
First, clear the fraction by multiplying the entire equation by 3:
⇒ 3 × (3x2) – 3 × (2x) + 3 × ( 1 3 ) = 3 × 0
⇒ 9x2 - 6x + 1 = 0

Comparing this equation with the standard form ax2 + bx + c = 0, we have:
a = 9
b = -6
c = 1

Now, calculate the discriminant (Δ):
⇒ Δ = b2 - 4ac
⇒ Δ = (-6)2 - 4(9)(1)
⇒ Δ = 36 - 36
⇒ Δ = 0

Determine the nature of the roots based on the value of Δ:
Since Δ = 0, the quadratic equation has two equal real roots.

Find the roots, as they are real:
⇒ x = -b ± √Δ 2a

⇒ x = -(-6) ± √0 2 × 9

⇒ x = 6 ± 0 18

⇒ x = 6 18

⇒ x = 1 3

Result:

The discriminant of the equation 3x2 – 2x + 1 3 = 0 is 0.
Since the discriminant is equal to zero (Δ = 0), the quadratic equation has two equal real roots.
The roots are x = 1 3 (repeated root).
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