SOLVING WORD PROBLEMS WITH QUADRATICS

Example 1 :

If a number is subtracted from its square, the result is 110. What is the number ?

Solution :

Let x be the number.

Its square is x².

Then,

x² – x  =  110

x² – x – 110  =  0

By factorization, we get

(x – 11) (x + 10)  =  0

x  =  11 and x  =  -10

we taking the positive value.

So, the number is 11.

Example 2 :

a)  Two numbers have a sum of 9. If one of them is x, what is the other number ?

b)  If the sum of the squares of the numbers in a is 45, find the numbers.

Solution :

a)

Given, one of the number is x

So, the other number is 9 - x

b)

Let the one of the number is x.

The other number is 9 - x

Its Squares are x² and (9 - x)²

x² + (9 - x)²  =  45

x² + 9² - 18x + x²  =  45

2x² - 18x + 81 – 45  =  0

2x² - 18x + 36  =  0

Dividing by 2, we get

x² - 9x + 18  =  0

By factorization, we get

(x – 6) (x – 3)  =  0

x  =  6 and x  =  3

So, the numbers are 6 and 3.

Example 3 :

A rectangle has length 3 cm greater than its width. If it has an area of 28 cm², find the dimensions of the rectangle.

Solution :

Let the width of the rectangle is x.

Length of the rectangle is x + 3.

Area  =  28cm²

Area of a rectangle  =  l × w

28  =  (x + 3)(x)

28  =  x² + 3x

x² + 3x – 28  =  0

By factorization, we get

(x – 4) (x + 7)  =  0

x  =  4 and x  =  -7

we taking positive value,

Now, width  =  4cm

Length  =  4 + 3

=  7cm

So, dimensions of the rectangle are 7cm × 4cm

Example 4 :

A rectangle enclosure is made from 38m of fencing. The area enclosed is 70 m². Find the dimensions of the enclosure.

Solution :

Let x be the length and y be the width.

Perimeter of a rectangle  =  2(l + b)

Here, P  =  38m

2(x + y)  =  38

2x + 2y  =  38 -----(1)

Area  =  70m²

xy  =  70

y  =  70/x -----(2)

By applying y  =  70/x in equation (1), we get

2x + 2(70/x)  =  38

2x + 140/x  =  38

2x² + 140  =  38x

2x² – 38x + 140  =  0

Dividing by 2, we get

x² – 19x + 70  =  0

(x – 5) (x – 14)  =  0

x  =  5 and x  =  14

By applying x  =  5 in equation (2), we get

y  =  70/x

=  70/5

y  =  14

Length  =  5m

Width  =  14m

So, the dimensions of the enclosure are 5m × 14m

Example 5 :

A triangle has a base which is 3cm longer than its altitude.

a)  Find its altitude if its area is 44 cm²

b)  Find its altitude if its area is 90 cm²

Solution :

a)

Let h be the altitude be.

Base is h + 3

Area  =  44cm²

Area of a triangle  =  1/2 × b × h

44  =  1/2 × (h + 3) × h

44  =  (h² + 3h)/2

88  =  h² + 3h

h² + 3h – 88  =  0

By factorization, we get

(h – 8) (h + 11)  =  0

h  =  8 and h  =  -11

we taking positive value.

So, the altitude is 8 cm.

b)

Let the altitude is h.

Base is h + 3

Area  =  90cm²

Area of a triangle  =  1/2 × b × h

90  =  1/2 × (h + 3) × h

90  =  (h² + 3h)/2

180  =  h² + 3h

h² + 3h – 180  =  0

By factorization, we get

(h – 12) (h + 15)  =  0

h  =  12 and h  =  -15

we taking positive value.

So, the altitude is 12cm.

Example 6  :

A small business makes surfboards and find that its profit, $P per hour, is given by the formula

P  =  75x – 5x²

where x is the number of surfboards made per hour. When does the business make :

a)  $0 profit per hour

b)  $250 profit per hour

Solution :

a)

P  =  $0

By the given formula,

P  =  75x – 5x²

0  =  75x – 5x²

5x² – 75x  =  0

5x(x – 15)  =  0

x – 15  =  0

x  =  15

So, 15 surfboards are made per hour.

b)

P  =  $250

By the given formula,

P  =  75x – 5x²

250  =  75x – 5x²

5x² – 75x + 250  =  0

Dividing by 5, we get

x² – 15x + 50  =  0

(x – 10) (x – 5)  =  0

x  =  10 or 5

So, 10 or 5 surfboards are made per hour.

Example 7  :

A manufacturer makes high quality ice-skates. The profit $P per day is given by the formula

P  =  72x – 3x² 

where x is the number of pairs of skates made per day. When does the manufacturer make :

a)  $420 profit per day

b)  no profit per day

Solution  :

a)

P  =  $420

By the given formula,

P  =  72x – 3x²

420  =  72x – 3x²

3x² – 72x + 420  =  0

Dividing by 3, we get

x² – 24x + 140  =  0

(x – 14) (x – 10)  =  0

x  =  14 or 10

So, 14 or 10 pairs of skates are made per day.

b)

P  =  $0

By the given formula,

P  =  72x – 3x²

0  =  72x – 3x²

3x² – 72x  =  0

3x(x – 24)  =  0

x – 24  =  0

x  =  24

So, 24 pairs of skates are made per day.

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