If a second degree equation can be factored, we will be able to rewrite the x-term as a sum of the factors and the constant as a product in the following way.
ccccccc
0&=&x^2 & + & - 13 & x & + & 40
0&=&x^2 & + & (a+b) & x & + & ab
Let's list all of the ways that 40 can be factored, and then investigate which pair of factors add to - 13. Notice that because the constant is positive and the x-coefficient is negative, both factors must be negative.
c|c|c
Integer & ab & a+b [0.3em]
40 & (- 1)(- 40) & - 41
40 & (- 2)(- 20) & - 22
40 & (- 4)(- 10) & - 14
40 & (- 5)(- 8) & - 13
From the table we see that if we substitute a= - 5 and b= - 8 into 0=(x+a)(x+b) we will have factored the equation.
0=(x+( - 5))(x+( - 8)) ⇔ 0=(x-5)(x-8)
When x=5 and x=8, the equation is solved. This tells us that when the rocket takes off it is 5 meters away from Robbie, and when it lands it is 8 meters away from Robbie.
b From Part A we know that the rocket takes off 5 meters away from the Robbie and that it lands 8 meters from Robbie. This must mean that when the rocket lands it is 8-5=3 meters away from the launch pad.
