Now solve each equation for x. You should come up with two solutions: x can
be either −^3 / 2 or 2.
Pro Tip: A particularly difficult math problem might ask for zeroes of a
polynomial. This just means that you need to set the expression equal to zero,
then solve for x. For the expression on the previous page (2x^2 – x – 6), the
question might be “What are the zeroes of the equation 2x^2 – x – 6 = 0?” To
solve it, just set the expression equal to zero (2x^2 – x – 6 = 0) and factor it (using
the method above) to find that x = – ^3 / 2 or 2. So our zeroes are – ^3 / 2 and 2. (It’s
just a mathier way of saying “solve for x.”)
All this might have taken you a minute. It might have taken you 20. But don’t
worry about it—the more problems you do, the better you get at seeing the
answer without having to do many calculations.
Not all trinomials can be factored like this, but more on that coming up.
Remember: Zero times anything equals zero. Therefore, if either factor equals zero, the whole expression must also be equal to zero.
—Samantha
Let’s try a problem:The sum of the two roots of a quadratic equation is 5 and their product
is –6. Which of the following could be the equation?
A) x^2 – 6x + 5 = 0
B) x^2 – 5x – 6 = 0
C) x^2 – 5x + 6 = 0
D) x^2 + 5x – 6 = 0
First of all, since the product is negative and the sum is positive, you know
that one of the numbers is negative and the other is positive. Start with the
product (there are fewer possibilities)—how many pairs of numbers would have
a product of –6? The answer: –1 and 6, –6 and 1, –3 and 2, –2 and 3. Then look
at the sum. Out of all these pairs, which one has a sum of 5? The pair –1 and 6
satisfies both conditions, so –1 and 6 are the roots.
Next, put those roots down as the answers to a quadratic equation—(x + 1) (x
–6) = 0. Then FOIL this out, and you’ll get the answer: B.
Pro Tip: In any quadratic equation ax^2 + bx + c = 0, the sum of the two roots
is equal to and the product of the two roots is equal to . Try it, it works!
QUADRATIC FORMULA