NUMBER WORD PROBLEMS WITH QUADRATIC EQUATIONS

Problem 1 :

The sum of a positive integer and its square is 90. Find the number.

Solution:   

Let x be an integer.

x² + x = 90

x² + x - 90 = 0

x² + 10x - 9x - 90 = 0

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

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

x = 9 or x = -10

So, an integer number is 9.

Problem 2 :

The sum of a positive integer and its square is 132. Find the number.

Solution:   

Let x be an integer.

x² + x = 132

x² + x - 132 = 0

x² + 12x - 11x - 132 = 0

x(x + 12) - 11(x + 12) = 0

(x - 11) (x + 12) = 0

x = 11 or x = -12

So, an integer number is 11.

Problem 3 : 

The difference between a positive integer and its square is 56. Find the number.

Solution:

Let x be an integer.

x² - x = 56

x² - x - 56 = 0

x² + 7x - 8x - 56 = 0

x(x + 7) - 8(x + 7) = 0

(x + 7) (x - 8) = 0

x = -7 or x = 8

So, an integer number is 8. 

Problem 4 :

The square of a number is equal to 5 times the number. What are the two possible answers?

Solution:

Let the number be x.

x² = 5x

x² - 5x = 0

x(x - 5) = 0

x = 0 or x = 5

So, the two possible numbers are 0 or 5.

Problem 5 :

When a number is subtracted from its square, the result is 42. Find the two possible solutions.

Solution:

Let the number be x.

x² - x = 42

x² - x - 42 = 0

x² - 7x + 6x - 42 = 0

x(x - 7) + 6(x - 7) = 0

(x + 6) (x - 7) = 0

x = -6 or x =7

So, the two possible numbers are -6 or 7.