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As we know that
Let the distance x and time taken be t.
Now According to the question,
And
From (1) we have
Putting this on (2), we get
Since speed can never be negative, we choose,
Hence the speed of the train is 40 km/hour.

L.H.S. ,
and R.H.S can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.

Let us assume the length and breadth of the park be respectively.
Then, the perimeter will be
The area of the park is:
Given :
Comparing to get the values of a, b and c.
The value of the discriminant
As .
Therefore, this equation will have two equal roots.
And hence the roots will be:
Therefore, the length of the park,
and breadth of the park .

Let the age of one friend be
and the age of another friend will be:
4 years ago, their ages were, and .
According to the question, the product of their ages in years was 48.
or
Now, comparing to get the values of .
Discriminant value
As .
Therefore, there are no real roots possible for this given equation and hence,
This situation is NOT possible.

Let the breadth of mango grove be .
Then the length of mango grove will be .
And the area will be:
Which will be equal to according to question.
Comparing to get the values of .
Finding the discriminant value:
Here,
Therefore, the equation will have real roots.
And hence finding the dimensions:
As negative value is not possible, hence the value of breadth of mango grove will be...

For two equal roots for the quadratic equation:
The value of the discriminant .
Given equation:
Can be written as:
Comparing and getting the values of a,b, and, c.
The value of
But is NOT possible because it will not satisfy the given equation.
Hence the only value of is 6 to get two equal roots.

For two equal roots for the quadratic equation:
The value of discriminant .
Given equation:
Comparing and getting the values of a,b, and, c.
The value of
Or,

The value of the discriminant
The discriminant > 0. Therefore the given quadratic equation has two distinct real root
roots are
So the roots are

Here the value of discriminant =0, which implies that roots exist and the roots are equal.
The roots are given by the formula
So the roots are

For a quadratic equation, the value of discriminant determines the nature of roots and is equal to:
If D>0 then roots are distinct and real.
If D<0 then no real roots.
If D= 0 then there exists two equal real roots.
Given the quadratic equation, .
Comparing with general to get the values of a,b,c.
Finding the discriminant:
Here D is negative hence there are no real roots possible for the...

Let the sides of the squares be . (NOTE: length are in meters)
And the perimeters will be: respectively.
Areas respectively.
It is given that,
.................................(1)
.................................(2)
Solving both equations:
or putting in equation (1), we obtain
Solving by the factorizing method:
Here the roots...

Let the average speed of the passenger train be .
Given the average speed of the express train
also given that the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance.
Therefore,
Can be written as quadratic form:
Roots are:
As the speed cannot be negative.
Therefore, the speed of the passenger train will be and
The speed...

Let the time taken by the smaller pipe to fill the tank be
Then, the time taken by the larger pipe will be: .
The fraction of the tank filled by a smaller pipe in 1 hour:
The fraction of the tank filled by the larger pipe in 1 hour.
Given that two water taps together can fill a tank in hours.
Therefore,
Making it a quadratic equation:
Hence the roots are
As the time taken cannot...

Let the speed of the train be
Then, time taken to cover will be:
According to the question,
Making it a quadratic equation.
Now, solving by the factorizing method:
However, the speed cannot be negative hence,
The speed of the train is .

Given the difference of squares of two numbers is 180.
Let the larger number be 'x' and the smaller number be 'y'.
Then, according to the question:
and
On solving these two equations:
Solving by the factorizing method:
As the negative value of x is not satisfied in the equation:
Hence, the larger number will be 18 and a smaller number can be found by,
putting x = 18, we...

Let the shorter side of the rectangle be x m.
Then, the larger side of the rectangle wil be .
Diagonal of the rectangle:
It is given that the diagonal of the rectangle is 60m more than the shorter side.
Therefore,
Solving by the factorizing method:
Hence, the roots are:
But the side cannot be negative.
Hence the length of the shorter side will be: 90 m
and the length of the larger...

Let the marks obtained in Mathematics be 'm' then, the marks obtain in English will be '30-m'.
Then according to the question:
Simplifying to get the quadratic equation:
Solving by the factorizing method:
We have two situations when,
The marks obtained in Mathematics is 12, then marks in English will be 30-12 = 18.
Or,
The marks obtained in Mathematics is 13, then marks in English will...

Let the present age of Rehman be years.
Then, 3 years ago, his age was years.
and 5 years later, his age will be years.
Then according to the question we have,
SImplifying it to get the quadratic equation:
Hence the roots are:
However, age cannot be negative
Therefore, Rehman is 7 years old in present.

Given equation:
So, simplifying it,
or
Can be written as:
Hence the roots of the given equation are:

Given equation:
So, simplifying it,
Comparing with the general form of the quadratic equation: , we get
Now, applying the quadratic formula to find the roots:
Therefore, the roots are

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