Important sums in areas related to quadratic equations

         (frequently asked in board exams)

Solve the following quadratic equation by factorisation method: x2 - 2ax + a2 - b2 = 0.

Sol. Here, Factors of constant term (a2 - b2) are (a - b) and (a + b).
Also, Coefficient of the middle term = - 2a = - [(a - b) + (a + b)]
\ x2 - 2ax + a2 - b2 = 0
x2 - {(a - b) + (a + b)}x + (a - b)(a + b) = 0
x2 - (a - b) x - (a + b) x + (a - b) (a + b) =
x{x - (a - b)}- (a + b) {x - (a - b)} = 0
{x - (a - b)} {x - (a + b)} = 0
x - (a - b) = 0 or, x - (a + b) = 0
x = a - b or x = a + b

Solve 64x2 - 625 = 0

Sol. We have 64x2 - 625 = 0
or (8x)2 - (25)2 = 0
or (8x + 25) (8x - 25) = 0
i.e. 8x + 25 = 0 o 8x - 25 = 0.
This gives .
x=-25/8,25/8
    

Solve the quadratic equation 16x2 - 24x = 0.

Sol. The given equation may be written as 8x(2x - 3) = 0 

x=0,3/2

Solve :- 25x2 - 30x + 9 = 0

Sol. 25x2 - 30x + 9 = 0 is equivalent to (5x)2 - 2(5x) × 3 + (3)2 = 0

or (5x - 3)2 =

This gives

     x=3/5,3/5

By Using Quadratic Formula :

Solve the quadratic equation in general form viz. ax2 + bx + c = 0.
We have, ax2 + bx + c = 0
Step (i) By comparison with general quadratic equation, find the value of a,b and c.
Step (ii) Find the discriminate of the quadratic equation.
D = b2 - 4ac
Step (iii) Now find the roots of the equation by given equation

APPLICATIONS OF QUADRATIC EQUATIONS :
ALGORITHM : The method of problem solving consist of the following three steps :

Step (i) Translating the word problem into symbolic language (mathematical statement) which means
identifying relationship existing in the problem and then forming the quadratic equation.
Step (ii) Solving the quadratic equation thus formed.
Step (iii) Interpreting the solution of the equation, which means translating the result of mathematical
statement into verbal language.

REMARKS :

Two consecutive odd natural numbers be 2x - 1, 2x + 1 where xÎN

Two consecutive even natural numbers be 2x, 2x + 2 where xÎN

Two consecutive even positive integers be 2x, 2x + 2 where + xÎZ

Consecutive multiples of 5 be 5x, 5x + 5, 5x + 10 .....

The sum of the squares of two consecutive positive integers is 545. Find the integers.

Sol. Let x be one of the positive integers. Then the other integer is x + 1, + xÎZ
Since the sum of the squares of the integers is 545, we get
x2 + (x + 1)2 = 545
or 2x2 + 2x - 544 = 0
or x2 + x - 272 = 0
x2 + 17x - 16x - 272 = 0
or x(x + 17) - 16(x + 17) = 0
or (x - 16) (x + 17) = 0
Here, x = 16 or x = -17 But, x is a positive integer. Therefore, reject x = - 17 and take x = 16. Hence, two
consecutive positive integers are 16 and (16 + 1), i.e., 16 and 17.

.15 The length of a hall is 5 m more than its breath. If the area of the floor of the hall is 84 m2, what are the
length and the breadth of the hall ?

Sol. Let the breadth of the hall be x metres.
Then the length of the ball is (x + 5) metres.
The area of the floor = x(x + 5) m2
Therefore, x(x + 5) = 84
or x2 + 5x - 84 = 0
or (x + 12) (x - 7) = 0
This given x = 7 or x = - 12.
Since, the breadth of the hall cannot be negative, we reject x = - 12 and take x = - only.
Thus, breadth of the hall = 7 metres, and length of the hall = (7 + 5), i.e., 12 metres.

.17 The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the
triangle is 5 cm. Find the lengths of these sides.

Sol. Let the length of the shorter side b x cm. Then, the length of the longer side = (x + 5) cm.
Since the triangle is right-angled, the sum of the squares of the sides must be equal to the square of the
hypotenuse (Pythagoras Theorem).
x2 + (x + 5)2 = 2352
or x2 + x2 + 10x + 25 = 625
or 2x2 + 10x - 600 = 0
or x2 + 5x - 300 = 0
or (x + 20) (x - 15) = 0
This gives x = 15 or x = - 20
We reject x = - 20 and take x = 15.
Thus, length of shorter side = 15 cm.
Length of longer side = (15 + 5) cm, i.e., 20 cm.

9 The sum of the square of two positive integers is 208. If the square of the larger number is 18 times the
smaller number, find the numbers. [CBSE - 2007]

Sol Let x be the smaller number.
Then, square of the larger number will be 18x.
Therefore, x2 + 18x = 208
or x2 + 18x - 208 =0
This gives x = 8 or x = - 26
Since the numbers are positive integers, we reject x = - 26 and take x = 8.
Therefore, square of larger number = 18 × 8 = 144.
So, larger number = 144 = 12
Hence, the larger number is 12 and the smaller is 8.

important questions for practice

OBJECTIVE 

1. If one root of 5x2 + 13x + k = 0 is reciprocal of the other then k =

2. The roots of the equation x2 - x - 3 = 0 are

(A) Imaginary (B) Rational (C) Irrational (D) None of these

3. The difference between two numbers is 5 different in their squares is 65. The larger number is

(A) 9 (B) 10 (C) 11 (D) 12

4. The sum of ages of a father and son is 45 years. Five years ago, the product of their ages was 4 times the age

of the father at that time. The present age of the father is

(A) 30 yrs (B) 31 yrs (C) 36 yrs (D) 41 yrs

5. If one of the roots of the quadratic equation is 2 + 3 then find the quadratic equation.

(A) x2 - (2 + 3 ) x+ 1 = 0 (B) x2 + (2 + 3 ) x + 1 = 0

(C) x2 - 4x + 1 = 0 (D) x2 + 4x - 1 = 0

SUBJECTIVE 

2. Find the value of k for which quadratic equation (k - 2)x2 + 2(2k - 3)x + 5k - 6 = 0 has equal roots.
3. The sum of the squares of two consecutive positive integers is 545. Find the integers.
4. A man is five times as old as his son and the sum of the squares of their ages is 2106. Find their ages.

5. The sides (in cm) of a right triangle containing the right angles are 5x and 3x - 1. If the area of the triangle is
60 cm2. Find its perimeter.
6. The lengths of the sides of right triangle are 5x + 2, 5x and 3x - 1. If x > 0 find the length of each sides.
7. A two digit number is four times the sum and three times the product of its digits, find the number
[CBSE – 2000]
8. The number of a fraction is 1 less than its denominator. If 3 is added to each of the numerator and
denominator, the fraction is increased by
28
3
. Find the fraction [CBSE - 2007]

10. An aeroplane left 30 minutes later then its scheduled time and in order to reach its destination 1500 km
away in time. it has to increase its speed by 250 km/h from its usual speed. Determine its usual speed.
[CBSE-2005]
11. A motor boat whose speed is 18 km/h in still water takes 1 hours more to go 24 km upstream than to return
downstream to the same spot. Find the speed of the stream. [CBSE-2008]
12. Two water taps together can fill a tank in 9 3/8
 hours. The tap of larger diameter takes 10 hours less that the
smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
[CBSE-2008]