WORD PROBLEMS ON CONSECUTIVE NUMBERS WITH QUADRATIC EQUATIONS

Problem 1 :

The product of two consecutive positive even numbers is 48. Find the numbers.

Solution:

Let the two consecutive even numbers are x and x + 2.

x(x + 2) = 48

x² + 2x = 48

x² + 2x - 48 = 0

x² + 8x - 6x - 48 = 0

x(x + 8) - 6(x + 8) = 0

(x - 6) (x + 8) = 0

x = 6 or x = -8 

Therefore, negative value is not possible.

x = 6 and x + 2 = 8

So, the two consecutive even positive numbers are 6 and 8.

Find the two positive integers required, if :

Problem 2 :

The numbers are consecutive and their product is 20.

Solution :

Let two consecutive numbers be x and x + 1.

x(x + 1) = 20

x² + x = 20

x² + x - 20 = 0

x² + 5x - 4x - 20 = 0

x(x + 5) - 4(x + 5) = 0

(x + 5) (x - 4) = 0

x = -5 or x = 4

Therefore negative value is not possible.

So, when x = 4,

Then x + 1 = 4 + 1 = 5

Thus, two consecutive numbers are 4, 5.

Problem 3 :

The numbers are consecutive and their product is 90.

Solution:

Let two consecutive numbers be x and x + 1.

x(x + 1) = 90

x² + x = 90

x² + x - 90 = 0

x² + 10x - 9x - 90 = 0

x(x + 10) - 9(x + 10) = 0

(x + 10) (x - 9) = 0

x = -10 or x = 9

Therefore negative value is not possible.

So, when x = 9,

Then x + 1 = 9 + 1 = 10

Thus, two consecutive numbers are 9, 10.

Problem 4 :

The numbers are consecutive even numbers and their product is 120.

Solution:

Let two consecutive even numbers be x and x + 2.

x(x + 2) = 120

x² + 2x = 120

x² + 2x - 120 = 0

x² + 12x - 10x - 120 = 0

x(x + 12) - 10(x + 12) = 0

(x + 12) (x - 10) = 0

x = -12 or x = 10

Therefore negative value is not possible.

So, when x = 10,

Then x + 2 = 10 + 2 = 12

So, the numbers are 10, 12.

Problem 5 :

The numbers are consecutive odd numbers and their product is 63.

Solution:

Let two consecutive odd numbers be x and x + 2.

x(x + 2) = 63

x² + 2x = 63

x² + 2x - 63 = 0

x² + 9x - 7x - 63 = 0

x(x + 9) - 7(x + 9) = 0

(x + 9) (x - 7) = 0

x = -9 or x = 7

Therefore negative value is not possible.

So, when x = 7,

Then 7 + 2 = 7 + 2 = 9

So, the numbers are 7, 9.