Question

Two consecutive odd integers have a product of 99 solve with a quadratic equation and find images

Answer 1

9 and 11

OR

-9 and -11

Explanation:

let x be the one odd integer

The pattern of numbers go: odd - even - odd - even

The next consecutive odd integer would be x + 2

"Product" means multiplying. Write "the product of two consecutive odd integers with a product of 99" as an algebraic statement:

x(x+2)=99 Distribute over brackets

x² + 2x = 99 Rearrange the equal 0

x² + 2x - 99 = 0

Remember a quadratic equation is ax² + bx + c = 0

(Equate to 0 to use quadratic formula)

State values for quadratic formula from simplified quadratic equation

a = 1; b = 2; c = -99

Use the quadratic formula.

`x=\frac{-b±\sqrt{b^(2)-4ac}}{2a}`

Substitute the values of "a", "b", and "c"

`x=\frac{-2±\sqrt{2^(2)-4(1)(-99)}}{2(1)}` Simplify

`x=(-2±√(400))/(2)` Solve the root

`x=(-2±20)/(2)`

Split the equation at the ±

`x=(-2+20)/(2)`

`x=(18)/(2)`

x = 9 Possible integer solution

`x=(-2-20)/(2)`

`x=(-22)/(2)`

x = -11 Possible integer solution

Integers include all positive and negative whole numbers, and 0. Both positive and negative answers are possible in this problem.

Use "x+2" to get the consecutive integer from the initial possible values for "x".

If x = 9:

x+2 = x+9 = 11

9 and 11

If x = -11:

x+2 = -11+2 = -9

-9 and -11

See if your answers make sense:

9 X 11 = 99

-9 X -11 = 99

Both are possible solutions.