Mathematics : Algebra: Solving Problems Involving Quadratic Equations: Steps to solve a problem

**Solving Problems Involving Quadratic Equations**

**Step 1 **Convert the word problem to a quadratic equation form

**Step 2 **Solve the quadratic equation obtained in any one of the above
three methods.

**Step 3 **Relate the mathematical solution obtained to the statement asked
in the question.

The product of Kumaran’s age (in years) two years ago and his age
four** **years from now is one
more than twice his present age. What is his present age?

Let the present age of Kumaran be* **x*** **years.

Two years ago, his age = (*x* − 2) years.

Four years from now, his age = (*x* + 4) years.

Given,

(*x* − 2)(*x* + 4) = 1 +2*x*

*x*^{2}* *+* *2*x *−* *8* *= 1 +2*x *gives* *(*x
*−* *3)(*x *+* *3)* *=* *0* *then,* x *= ±3

Therefore, *x* = 3 (Rejecting −3 as age cannot be negative)

Kumaran’s present age is 3 years.

A ladder** **17** **feet long is leaning against a wall. If the ladder, vertical wall** **and the floor from the
bottom of the wall to the ladder form a right triangle, find the height of the
wall where the top of the ladder meets if the distance between bottom of the
wall to bottom of the ladder is 7 feet less than the height of the wall?

Let the height of the wall* **AB*** **=

As per the given data *BC* = (*x*–7) feet

In the right triangle *ABC*, *AC* =17 ft, *BC* = (*x*–7)
feet

By Pythagoras theorem, *AC*^{2} = *AB*^{2}
+ *BC*^{2}

(17)^{2} = *x*^{2} + (*x* −
7)^{2} ; 289 = *x*^{2} + *x*^{2} − 14*x*
+ 49

*x*^{2}* *−* *7*x *−120* *=* *0* *hence,* *(*x
*−* *15)(*x *+* *8)* *=* *0* *then,* x *=*
*15* *(or) −8

Therefore, height of the wall AB = 15 ft (Rejecting −8 as height cannot be negative)

A flock of swans contained** ***x*^{2}** **members. As the clouds gathered,** **10*x*** **went** **to a lake and one-eighth
of the members flew away to a garden. The remaining three pairs played about in
the water. How many swans were there in total?

As given there are* **x*^{2}** **swans.

As per the given data *x*^{2} − 10*x* – (1/8)*x*^{2} = 6 we get, 7*x*^{2}
− 680*x* − 48 = 0

Therefore, *x *= 12, -4/7

Here *x *= 4/7 is not possible as the number of swans cannot
be negative.

Hence, *x* = 12. Therefore total number of swans is *x*^{2}
= 144.

A passenger train takes** **1** **hr more than an express train to travel a
distance**
**of 240 km from Chennai
to Virudhachalam. The speed of passenger train is less than that of an express
train by 20 km per hour. Find the average speed of both the trains.

Let the average speed of passenger train be* **x*** **km/hr.

Then the average speed of express train will be (*x* + 20)
km/hr

Time taken by the passenger train to cover distance of 240 km = 240/*x* hr

Time taken by express train to cover distance of 240 km = 240 / (*x*+20) hr

Given,

*x *^{2}* *+* *20*x *–* *4800 = 0 gives, (*x* + 80)(*x*
− 60) = 0 we get, *x* = –80 or 60.

Therefore *x* = 60 (Rejecting -80 as speed cannot be
negative)

Average speed of the passenger train is 60 km/hr

Average speed of the express train is 80 km/hr.

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