Respuesta :
The equation in terms of x is x(x + 2) = 99 or [tex]x^2 + 2x - 99 = 0[/tex]
The pairs of integers are -11 and -9 or 9 and 11
Solution:
Let the two consecutive odd integers be x and x + 2
Let "x" be the smallest integer
Given that product of two consecutive odd integers is 99
To find: an equation in terms of x
Product of x and x + 2 = 99
[tex]x(x + 2) = 99\\\\x^2 + 2x = 99\\\\x^2 + 2x - 99 = 0[/tex]
Thus the equation in terms of x is found
Let us solve the above equation to find all pairs of integers
Solve the quadratic equation by grouping method
Middle term 2x can be written as 11x - 9x
[tex]\rightarrow x^2 + 2x - 99 = 0\\\\\rightarrow x^2 + 11x - 9x -99 = 0[/tex]
Now group the terms,
[tex]\rightarrow (x^2 + 11x) - (9x + 99) = 0[/tex]
Take "x" as common term from first group and 9 as common term from second group
[tex]\rightarrow x(x + 11)-9(x+ 11) = 0[/tex]
Now take x + 11 as common term,
[tex]\rightarrow (x + 11)(x - 9) = 0[/tex]
Equating to zero we get,
x + 11 = 0 or x - 9 = 0
x = -11 or x = 9
When x = -11 :
x + 2 = -11 + 2 = -9
Thus first set of two consecutive odd integers are -11 and -9
When x = 9 :
x + 2 = 9 + 2 = 11
Thus second set of two consecutive odd integers are 9 and 11
