Question

chapter - quadratic equations
topic - roots and nature of roots
class - 11th

i want explanation in detail pls, thx in advance if you did it right :)
go for the brainliest answer!!!
​

Answer 1

Answer:

(4) Imaginary

Step-by-step explanation:

We are given that the roots of the quadratic equation ax² + x + b = 0 are real and distinct and we are asked to find the nature of roots of the equation x² - 4√(ab)x + 1 = 0.

In order to solve this problem, we should know about the nature of roots on the basis of value of discriminant.

Discriminant, or in short represented as D, can be obtained by using following formula :-

D = b² - 4ac

Here,

• D = discriminant
• a = coefficient of x² in equation
• b = coefficient of x in equation
• c = constant term in equation

Nature of roots is characterized as,

• If D = 0 implies that roots are real and equal to each other.
• If D > 0 ( Greater than 0 ) implies that roots are real and distinct from each other.
• If D < 0 ( Smaller than 0 ) implies that roots are not real but imaginary.

Since the given equation, ax² + x + b = 0 has real and distinct roots, value of D ( Discriminant ) will be greater than 0.

→ D = ( coefficient of x )² - 4 ( coefficient of x²) ( constant term)

→ 0 < ( 1 )² - 4 ( a ) ( b )

→ 0 < 1 - 4 ab

→ 4 ab < 1

→ 4ab - 1 < 0 --- Eqⁿ(1)

Now we will find nature of roots of x² - 4√(ab) x + 1 = 0.

In the equation,

• Coefficient of x = -4 √(ab)
• Coefficient of x² = 1
• Constant term = 1

Applying the formula of discriminant, we get :

→ D = b² - 4ac

→ D = (-4√ab)² - 4 (1) (1)

→ D = 16 ab - 4

→ D = 4 ( 4ab - 1 )

Now since, value of ( 4 ab - 1 ) is smaller than 0, or negative ( from Eqⁿ(1) ), when multiplied by 4 will also give a negative number. So the value of D will always be negative for this equation D < 0.

Hence the given equation has imaginary roots ( or no real roots ).