SAT Tip: Solving Quadratic Equations
This tip on improving your SAT score was provided by Vivian Kerr at Veritas Prep.
To factor means to break down a number into two smaller parts. For example, we know 12 = 3 x 4, so we can say that 3 and 4 are factors of 12, because they divide evenly into 12. We’d call 12 a multiple of both 3 and 4.
A polynomial is a math expression with multiple terms, such as z + 2y. It can consist of numbers and variables. To factor a polynomial, we just divide out what is common to all terms. We couldn’t factor the polynomial z + 2y because the two terms have nothing in common. But we could factor 6x + 7xy since both terms contain an“x.” If we divided the “x” out from both terms, we’d be left with 6 + 7y. The factors would look like this: (x)(6 + 7y).
If you need to check your work and are not sure if you factored correctly, you can always multiply the factors to see if you arrive at the original polynomial.
A quadratic equation is a special polynomial with three terms in the form ax² + bx + c. Why do quadratics look like this? Because of FOIL!
FOIL stands for First-Outer-Inner-Last. When we multiply the first two terms of the factors, the outer two terms, the inner two terms, and the last two terms, then sum them together, we’ll always arrive at the form ax² + bx + c! Let’s try it!
Say (x + 1)(x – 5) are the factors. The first two terms are (x)(x) = x2. The outer two terms are (x)(-5) = -5x. The inner two terms are (1)(x) = x. The last two terms are (1)(-5) = -5. Summed together we get: x2 + x -5x – 5 = x2 -4x -5. We can use this knowledge to help us factor.
1. Factor x² – 9x + 18.
Let’s start by setting up two parentheses: ( )( ).
The first term in each parentheses will always be x, since x*x = x². Then we look at the coefficient of the second term, -9. The second terms in the factors must add together to equal the middle term’s coefficient.
This means we need two numbers that sum to -9. There are quite a few combinations that would work: -1 and -8, -2 and -7, etc. But we know the third term in a quadratic will equal the product of the second terms in its factors, so the only two numbers that work are -6 and -3. Therefore the factors must be: (x – 3) (x – 6).
To find the roots or solutions of a quadratic equation, set the two factors each equal to zero and solve.
(x – 3) = 0 (x – 6) = 0
+3 +3 +6 +6
x = 3 x = 6
The “solutions” to this quadratic are 3 and 6. Remember on SAT test day that the “solution” to a question asking for the factors will be the factors, but the “solutions” to a quadratic are different.
Vivian Kerr has been teaching and tutoring in the Los Angeles area since 2005. She graduated from the University of Southern California, studied abroad in London, and has worked for several test-prep giants tutoring, writing content, and blogging about all things SAT, ACT, GRE, and GMAT.
For more SAT advice from Veritas Prep watch “SAT Tip: What Should You Do When You’re Stuck On a Question?”