4.5 Basic Factoring
The only way for a product to equal 0 would be if at least one of its factors is 0. Therefore, if we can rewrite an equation as a product, such as
then we can solve the equation by setting each of the factors equal to 0.
Continuing from the last section, the first question that we'd like to answer is "When is our egg at a height of 1.7 meters?", which algebraically translates to solving the equation
-4.9x^2+2.45x+1.7=1.7
To do so, let's get all of the x's to one side by subtracting 1.7 from each side. We're now left with
-4.9x^2+2.45x=0
If you stare at the equation long enough, one of the solutions may pop out at you. Without doing any algebra, one of the answers is when x = (Don't be afraid, give it a shot!). But is this the only solution? Since our function is an upside down quadratic we should realize that there very well might be another solution out there: what goes up must come down.
We could try to guess what the other solution is. However, by using the powerful distribution property, the other solution will pop out at us. Let's make use of the property that for any real numbers a, b, and c:
ab+ac=a(b+c)
If we let a=x, \; b=-4.9x, \; c=2.45, then we can rewrite the left-hand side of our original equation as:
-4.9x^2+2.45x=x(-4.9x) + x(2.45)=x(-4.9x+2.45)
You can think of this process as "pulling out the x"; we pulled an x term out of each of the pieces on the left-hand side. As a result, we transformed the original equation from being the sum of two things to being the product of two things. How does this help us? Well, another property of the real numbers is that:
ab= 0 \rm if\; and\; only\; if a=0 \rm \;or \ b=0
In other words, if the product of two numbers is equal to 0 then at least one of those numbers has to be 0.
Then starting with our rewritten equation, and setting it back equal to 0, we get that
x(-4.9x+2.45)=0
We then see that either x=0 or The second equation is just a linear equation. We can now use an old tool to solve a new problem: a powerful problem solving technique used in mathematics. Subtracting 2.45 from both sides and then dividing both sides by -4.9 we get that x=
.
We now see that the two solutions are:
x=0{\rm\; or \;}x=.5
The answer to our first question is that the egg is at a height a height of 1.7 meters when 0 seconds have passed and when .5 seconds have passed.
The process of "reverse distribution" is known as factoring, and it is an extremely powerful technique that we use to find the solutions to many different types of equations. If an equation can be rewritten as "product of stuff = 0", then you can just look at the individual factors and set each one equal to 0. Using addition to find solutions is difficult, but using multiplication is not!
Explore!
The two factors of -t^3+t^2 are
Can we use this powerful new factoring method to answer our second question, namely "When does the egg hit the ground?" If we want to know when the egg hits the ground, we're really trying to solve the equation -4.9x^{2}+2.45x+1.7= . The only technique that we have so far to solve quadratic equations is to try and factor the equation.
Can you factor the left-hand side of the equation above by pulling out an x?
Unfortunately we're stuck. How is this equation different from, say, -4.9x^{2}+2.45x=0?
In this case, we have the constant term 1.7 added as well, so we can't just "pull out an x". Might there be another way to factor the expression on the left-hand side? We have already seen that x(x + number) won't work; but what about something more interesting such as
(x + number)(x + number)?
We'll begin to explore this possibility in the next section.
