# Problems on Quadratic Equations

### General suggestions for solving applied problems:

- Read Statement of the problem carefully and determine what quantity (or quantities) are needed.
- Represent the unknown quantity (or quantities) by a letter (or letters).
- Determine the expressions which are equal and write an equation (or equations).
- Solve the resulting equation (or equations).

## Exercise

- Find two positive consecutive numbers such that the sum of their squares is 85.
- If the product of two positive consecutive odd integers is 195, find the integers.
- The sum of two numbers is 18 and the sum of their squares is 170. Taking one number as x, form an equation and solve it to find the numbers.
- The sum of the numerator and denominator of a certain positive fraction is 11. If 1 is added to both numerator and denominator, the fraction is increased by 3/56. Find the fraction.
- Two squares have sides (x +6) cm and (2x +1) cm. The sum of their areas is 697 sq. cm. Express this as an algebraic expression and solve the equation to find the areas of the squares.
- A rectangle of area 105 cm² has its length equal to x cm. Write its breadth in terms of x. If the perimeter is 44 cm, write an equation in x and solve it to find the dimensions of the rectangle.
- The length of a rectangle is 3 cm more than its width. If the length is increased by 1 cm and the width is increased by 3 cm, the area is double the area of the original rectangle. Find the dimensions of the original rectangle.
- A rectangular garden 10 m by 16 m is to be surrounded by a concrete walk of uniform width. Given that the area of the walk is 120 sq. meters, assuming the width of the walk to be x meters, form an equation in x and solve it to find the width of the walk.
- The perimeter of a rectangular plot is 68 m and length of its diagonal is 26 m. Find its area.
- A 40 cm long wire is bent to form a right-angled triangle with hypotenuse 17 cm. Find the area of the triangle so formed.
- In an auditorium, the number of rows was equal to the number of seats in each row. If the number of rows is increased by 6 and the number of seats in each row is increased by 2, then the total number of seats is increased by 172. How many rows were there?
- An express train makes a run of 240 km at a certain speed. Another train, whose speed is 12 km/hr less, takes an hour longer to make the same trip. Find the speed of the express train.
- A train covers a distance of 600 km at x km/hr. Had the speed been (x +20) km/hr, the time taken to cover the distance would have been reduced by 5 hours. Write down an equation in x and solve it to evaluate x.
- An aeroplane flying with a wind of 30 km/hr takes 40 minutes less to fly 3600 km, than what it would have taken to fly against the same wind. Find the planes speed of flying in still air.
- A swimming pool can be filled by 2 pipes together in 6 hours. If the larger pipe alone takes 5 hours less than the smaller pipe to fill the pool, find the time in which each pipe alone would fill the pool.
- Three numbers are in the ratio 1/15:1/10:1/6. If the sum of their squares is 152, find the numbers.
- When the cost price of an article is reduced by Rs 3, ten more articles can be bought for Rs 360. Find the original price of each article.
- The hotel bill for a number of people for overnight stay is Rs 4800. If there were 4 more, the bill each person had to pay would have reduced by Rs 200. Find the number of people staying overnight.
- A trader bought a number of articles for Rs 900. Five articles were damaged and he sold each of the rest at Rs 3 more than what he paid for it, thus getting a profit of Rs 150 on the whole transaction. Find the number of articles he bought.
- Peter bought a watch for Rs 70x and sold it for Rs (750 +2x) at a profit of x%. Find the cost price of the watch.
- A two digit number contains the bigger at unit place. The product of the digits is 24. If the difference between the digits is 5, find the number.
- A year ago, father was 8 times as old as his son. Now his age is the square of his sons age. Find their present ages.
- Two years ago, a mans age was three times the square of his sons age. In three years time, his age will be four times his sons age. Find their present ages.
- The length (in cm) of the hypotenuse of a right-angled triangle exceeds the length of one side by 2 cm and exceeds twice the length of other side by 1 cm. Find the length of each side.
- A speedboat travelled downstream at a distance of 10 km at an average speed of v km/hr. On the return journey, the average speed was increased by 5 km/hr and the journey took 6 minutes less. Find v.
- An aeroplane travelled a distance of 400 km at an average speed of x km/hr. Write down an expression for the time taken. On the return journey, the speed was increased by 40 km/hr. Write down an expression for the time for the return journey. If the return journey took 30 minutes less than the outward journey, write down an equation in x and solve it.
- A target on a dartboard has a center region of radius R cm and a concentric circle surrounding it of radius (R +1) cm. If the area between the two circles is one-tenth of the area of the center region, find R.

## Answers

**1.**6, 7

**2.**13, 15

**3.**7, 11

**4.**4/7

**5.**(x +6)² +(2x +1)² = 697; 256 sq. cm, 441 sq. cm.

**6.**cm; = 44; 15 cm, 7 cm

**7.**length = 7 cm, breadth = 4 cm

**8.**(16 + 2x)(10 +2x) -16.10 = 120; 2 m

**9.**240 m².

**10.**60 cm²

**11.**20

**12.**60 km/hr

**13.**600/x - 600/(x+20)= 5; x = 40

**14.**570 km/hr

**15.**10 hours, 15 hours

**16.**4, 6, 10

**17.**Rs 12

**18.**8

**19.**7

**20.**Rs 700

**21.**38

**22.**7 years, 49 years

**23.**5 years, 29 years

**24.**15 cm, 8 cm, 17 cm

**25.**20

**26.**400/x hrs; 400/(x+40) hrs; 400/(x+40)= 400/x - 1/2; x =160

**27.**20·5 cm approximately