Question

The product of two consecutive integers is 72. F ind all solutions.

Answer 1

x = -9 0r -8

x= 8 0r 9

Explanation:

let the two consecutive integers be

x

and x+ 1

so, the product of the two integers that is equal to 72 will be

x ( x+ 1) = 72

x² + x = 72

x² + x - 72 =0

note this expression x² + x - 72 =0 also looks like a quadratic equation

ax² + bx + c = 0

using the quadratic formula

x = -b±√b² - 4ac/2a

where b a = 1

b = 1

c = -72

x = -b± √ 1² - 4(1) ×(-72)/2(1)

x = -1 ± √ 1 + 288/2

x = -1± √289/2

x = -1 ± 17/2

x = -1-17/2 or -1 + 17/2

x = -18/2 0r 16/2

x = -9 or x = 8

so we have four answers that satisfy the initial equation

from

either

x= -9 or (x +1)= -9 + 1= -8

-9 x -8

72

also x =8

0r (x + 1) = 8+1 = 9

that is, 8 x 9 = 72

Answer 2

Case1: 8,9

Case 2 -9,-8

Explanation:

Let x = 1st integer

(x+1)= consecutive integer

x(x+1) = 72

Distribute

x^2 +x = 72

Subtract 72 from each side

x^2 +x -72 =0

Factor

What numbers multiply to -72 and add to 1

-8 *9 = -72

-8+9 = 1

(x-8) (x+9) =0

Using the zero product property

x-8 =0 x+9=0

x =8 x = -9

Case 1

x=8

x+1 =9

Case2

x = -9

x+1 = -8