5.5 SAT Math - Quadratic Equations

Quadratic equations on the SAT test are not that difficult.  You do NOT need to know the quadratic formula, and in fact, using it may often be less helpful than not because of one simple strategy:
-SAT Strategy:  Do the Opposite to Quadratic Equations.  If you see a quadratic equation on this test, factor it.  If you see a factored quadratic equation, expand it to standard form.  Almost every quadratic equation on this test will be solved very simply by following this rule!  Don't worry about if it looks weird or you don't understand why you're doing it, just put it in the opposite form from how it was presented to you, and see where that gets you.

Factored and Expanded Quadratic Equations

What do factored and expanded quadratic equations look like?  Quadratic equations in standard or expanded form look like ax² + bx + c, where a ,b and c are all coefficients, usually numbers.  So some examples are 2x² + 4x + 8 or x² - 5x - 6.  Sometimes, the SAT testmakers can try to trick you by giving you something like 4x² = 6x + 12, but if you subtract the numbers on the right side of the equals sign by both sides, you'll end up with a regular ol' quadratic equation.  The other tricky type looks like x² - y², where either of those variables could be numbers.  This one doesn't have a middle value, but it's still a quadratic equation.  So basically you want to look for one or more squared variables separated from other terms by a plus or minus sign. 

So what are the factored and expanded versions?  The most common ones you'll see on this test look like this:

If you've done quadratic equations in school, you may not need to memorize the first two as it may be pretty easy to figure them out as you go.  The third form, however, is called "the difference of two squares" and it's important to have memorized how to factor it because it's usually the only way to solve those types of quadratic equations on this test.  There are, of course, quadratic equations that don't fit one of the above simple versions, and for those use FOIL to solve:
First Outer Inner Last

First Outer Inner Last (FOIL)

To use FOIL, multiply the First two terms in a pair of parentheses together, then the Outer terms, then the Inner, then the Last:

To factor an expanded equation, such as x² - 5x + 6, begin by setting up two pairs of parentheses:

The front two terms should multiply out to the first term of the expanded equation; the back two numbers should multiply out to the last term of the expanded equation, so in this case we need to get x² in the front and positive 6 in the back.  Let's try 3 and 2 as two numbers that will multiply out to 6:

Does this work?  Use FOIL to check:

The front and back terms work, but the middle will add up to 5x.  We need it to add up to -5x.  So let's try making 3 and 2 negative:

It worked. 
-SAT Math Tip: Always be sure to double check the work you did by using FOIL.  If you get the signs wrong in your parenthetical factors, those answers will be available as answer choices, and you'll get the whole thing wrong.

Each of those expressions in parentheses is considered a factor of the equation.  How do you solve the equation?  It needs to be made equal to zero before you begin to factor.  So if you're given x² + 8x = 9, you first need to subtract 9 from both sides to get the equation equal to zero.
-SAT Math Hint: Be careful!  The above equation may appear to be one in which you should factor out an x rather than set it equal to zero.  Depending on what's asked, that may be true, but it's far more likely they're testing you on quadratic equations, and in that case, you always need to set it to zero.  If you have something squared, work with it according to quadratic rules rather than pulling out numbers in common.

Once you have x² + 8x - 9 = 0 you can factor the quadratic.  Let's try 3 and -3 to get -9:

(x+ 3)(x- 3)

Does it work?  Check it with FOIL.  No, it doesn't.  We get x² - 9 when we work it back out.  What else can we try?  Don't forget about the number itself.  Let's try 9 and -1:
(x + 9)(x - 1)
Does it work?  Yes.  Now what?  Because anything times zero equals zero, and we know that the whole equation when multiplied out equals zero, we know that one of these two factors must equal zero.  So take each of those parenthetical fractions and set it equal to zero to get your possible answers:

The answers to the equation x² + 8x - 9 = 0 is x=-9 or x=1.  These two answers are called the roots of the equation, or more often, a solution set.  They may appear in the answer choices as {-9, 1}.  Notice the curvy brackets?  Those refer to a set.  Parentheses around two numbers with a comma in the center refer to an ordered pair, something we'll talk about when we cover Geometry.
-Trick to Watch Out For: Quadratic equations will always have two answers, but sometimes the question itself will give you limits.  For example, using the same equation as above, the test question may read: "If x² + 8x = 9 and x≥0, what is the value of x?"  In this question, the values of x according to math rules are -9 and 1, but 1 will be the only valid answer because the question specified they only wanted the values of x greater than or equal to zero.  Also, keep in mind that on certain equations, both values of x might work out to be the same number.  That's okay.  That's just your answer.

Keep in mind the Trick from earlier in the Algebra section: it will almost always be easier to solve for the expression asked for than for each variable separately!  If you remember that Trick as well as to always put a quadratic equation in the opposite form in which they've given it to you, almost every quadratic equation on this test is extremely simple.  The way quadratics tend to be tested is like this:

If x² + 2xy+ y² = 25, what is x + y?

This is a long, perhaps seemingly impossible problem to solve if you don't remember to factor, or if you try to solve for x, then solve for y, then add the two together.  If you remember the rules, however, you'll quickly factor x² + 2xy+ y² into (x + y)², and all of a sudden you're facing the problem (x + y)² = 25, and they're just asking for x + y.  Which means all you have to do is take the square root of both sides, get 5 for an answer, and move on.  Very quick, very simple. 
-SAT Math Hint: If a quadratic equation is taking you forever, particularly if they're asking you for an expression like x + y or xy, stop.  Ask yourself, "did I factor this?"  Or if it's factored already, "did I FOIL this to expand?"  That's probably all you'll need to do to quickly find an answer.