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**Q.1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them;****(i) 2 x^{2} – 3x + 5 = 0**

(i) Given,

Comparing the equation with

We know, Discriminant =

= – 31

As you can see, b

Therefore, no real root is possible for the given equation,

(ii) 3

Comparing the equation with

We know, Discriminant =

= (-4√3)

= 48 – 48 = 0

As

Real roots exist for the given equation and they are equal to each other.

Hence the roots will be –

–

Therefore, the roots are 2/√3 and 2/√3.

(iii) 2

Comparing the equation with

As we know, Discriminant =

= (-6)

= 36 – 24 = 12

As

Therefore, there are distinct real roots exist for this equation, 2

= (-(-6) ± √(-6

= (6±2√3 )/4

= (3±√3)/2

Therefore the roots for the given equation are (3+√3)/2 and (3-√3)/2

(i) 2

Comparing the given equation with

As we know, Discriminant =

= (

=

For equal roots, we know,

Discriminant = 0

k = ±√24 = ±2√6

(ii)

or

Comparing the given equation with

We know, Discriminant =

= ( – 2

= 4

For equal roots, we know,

4

4

Either 4

However, if

Therefore, if this equation has two equal roots,

Let the breadth of mango grove be

Length of mango grove will be 2

Area of mango grove = (2

2

Comparing the given equation with

As we know, Discriminant =

=> (0)

Here,

Thus, the equation will have real roots. And hence, the desired rectangular mango grove can be designed.

As we know, the value of length cannot be negative.

Therefore, breadth of mango grove = 20 m

Length of mango grove = 2 × 20 = 40 m

Let’s say, the age of one friend be x years.

Then, the age of the other friend will be (20 – x) years.

Four years ago,

Age of First friend = (

Age of Second friend = (20 –

As per the given question, we can write,

(

16

Comparing the equation with

Discriminant =

= 400 – 448 = -48

Therefore, there will be no real solution possible for the equations. Hence, condition doesn’t exist.

Let the length and breadth of the park be

Perimeter of the rectangular park = 2 (

Or,

Area of the rectangular park =

Comparing the equation with

Since, Discriminant =

= 1600 – 1600 = 0

Thus,

Therefore, this equation has equal real roots. Hence, the situation is possible.

Root of the equation,

Therefore, length of rectangular park,

And breadth of the park,

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