TSI Math Study Guide
Quadratics and Polynomials
Quadratic — a polynomial whose highest exponent is 2, such as 3x2 − 5x + 2.
On the TSI you may need to simplify polynomial expressions, factor polynomials, solve quadratic equations, and evaluate quadratic functions.
Combining like terms
Multiplying polynomials
Monomial × polynomial
Distribute to each term.
Binomial × binomial
Distribute each term of the first.
Some students remember this as FOIL. The distributive property works in every case and is more general.
Factoring polynomials
Greatest common factor first
Before anything else, check whether every term shares a factor.
6x2 + 18x = 6x(x + 3)
Factoring trinomials
x2 + bx + c → find two numbers that multiply to c and add to b
Factor x2 + 7x + 12
Multiply to 12, add to 7 → 3 and 4(x + 3)(x + 4)
Factor x2 − x − 12
Multiply to −12, add to −1 → −4 and 3(x − 4)(x + 3)
Difference of squares
a2 − b2 = (a − b)(a + b)
Solving quadratic equations by factoring
First factor
x + 3 = 0
x = −3
Second factor
x + 4 = 0
x = −4
A quadratic can have two solutions because two different x-values may make the equation true.
Set it equal to zero first
The zero-product property only applies when the product equals zero. Factoring x2 + 2x = 15 into x(x + 2) = 15 tells you nothing.
The quadratic formula
Not every quadratic factors easily. The formula works for any ax2 + bx + c = 0.
With +
x = −1 + 2
With −
x = −1 − 2
Factoring or the formula?
Factors quickly → factor
x2 − 9x + 20 = 0 becomes (x − 4)(x − 5) = 0, so x = 4 or 5. No reason to use the formula.
Doesn’t factor easily → formula
The formula always works. The TSI may test whether you recognize which method is efficient, not just whether you can do the calculation.
Evaluating quadratic functions
For f(x) = ax2 + bx + c, substitute the given value for x and follow the order of operations.
f(x) = 2x2 − 3x + 4, find f(3)
f(x) = x2 + 2x, find f(−3)
When substituting a negative number, put it in parentheses. (−3)2 = 9, but −32 = −9.
Recognizing a quadratic relationship
A quadratic function doesn’t have a constant rate of change like a linear one. Its graph is a parabola — U-shaped. A positive x2 coefficient opens upward; a negative one opens downward.
You don’t need to treat every quadratic graph as its own topic, but knowing the basic shape helps with questions that compare types of functions.
TSI strategy: factor before reaching for the formula
Write the equation as ax2 + bx + c = 0, then look for an easy factorization.
Ask
Can I find two numbers that multiply to c and add to b? If yes, factor. If not, the formula always works.
Quadratic equations often have two solutions. Don’t stop after finding one.
Getting the equation equal to zero is usually the key first step. Once you have ax2 + bx + c = 0, you can factor or use the formula.
And when factoring gives (x − r)(x − s) = 0, set each factor equal to zero. That’s how you find all the solutions.