Let the unknown value be 'x'
Consider the three consecutive multiples of 5 as 'x', '(x + 5)' and '(x + 10)'
First number = x
Second number = (x + 5)
Third number = (x + 10)
Using the above statement, "The square of the third number, decreased by 5 times the second number = 25 more than twice the product of the first two numbers," let us form an equation
(x + 10)² - 5(x + 5) = 25 + (2 (x) (x + 5) )
x² + 20x + 100 - 5x - 25 = 25 + 2x² + 10x
x² + 15x + 75 = 2x² + 10x + 25
Bringing everything to one side,
0 = 2x² + 10x + 25 -x² -15x -75
or
2x² + 10x + 25 -x² -15x -75 = 0
x² - 5x - 50 = 0
Factorising the above equation,
(x - 10) (x + 5) = 0 ⇒ x = 10 or x = -5
If x = 10, then the three consecutive numbers are 10, 15, 20.
Let us check if it follows the above statement!
Square of the third number, decreased by 5 times the second number = 25 more than twice the product of the first two numbers
20² - (5 x 15) = 25 + (2 x (10 x 15))
400 - 75 = 25 + (2 x 150)
325 = 25 + 300
325 = 325
Now let us try with x = -5. The three consecutive numbers are -5, 0, 5
Let us check if it follows the above statement.
Square of the third number, decreased by 5 times the second number = 25 more than twice the product of the first two numbers
5² - (5 x 0) = 25 + (2 x (-5 x 0))
25 - 0 = 25 + (2 x 0)
25 = 25 + 0
25 = 25
Hence both x = 10 and x = -5 are correct
