SAT Math: Completing the Square

When the new SAT came out in March 2016, it became clear that the test makers were looking to test some skills that hadn’t appeared on the previous version of the test. One of those skills is “completing the square,” which can be helpful when you are working on problems involving circles and parabolas.

So what is completing the square? Well, a perfect square trinomial x² + bx + c is one that can be rewritten in the form (x + k)², where k is some integer. For instance x² + 10x + 25 is a perfect square trinomial because is can be factored into (x + 5)(x + 5) = (x + 5)².

When you are asked to complete the square, you need to find that value of c (there is always only one possible value) that will make the trinomial a perfect square. For instance, suppose you were asked this question:

What number will complete the square for x²  + 6x + ___ ?

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SAT Math: Algebraic Operations

Now more than ever, with the focus of the New SAT Math test being on algebra and functions, you need to make sure that you are an ace at algebraic operations. This includes things like simplifying algebraic expressions, solving linear and quadratic equations, and solving for a variable in a formula.

At the bottom of the page, we have links to 10 SAT practice problems that involve these topics, along with their solutions.

But first, a few friendly reminders (things you’ll probably remember your algebra teacher saying many times over!):

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SAT Math: Quadratics & Parabolas

The focus of the new SAT Math test is on concepts of algebra more than it has been in the past. One of the things you’ll want to know well is how to work with quadratic equations and their graphs, which are parabolas. You’ll need to be able to solve quadratic equations by factoring and by using the Quadratic Formula. You’ll also need to know about the different forms of the equation for a parabola and about the properties of parabolas.

At the bottom of this post you’ll find a link to ten SAT practice problems involving quadratics and parabolas. There’s a second link down there to the solutions as well.

But first, let’s review some of the important characteristics of quadratic functions and their graphs.

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SAT Math: Systems of Equations

The new SAT has shifted the focus of the math questions away from geometry and more toward algebra and functions. One of the topics you’ll want to master is systems of linear equations. If the practice tests are any indication, you can expect to see four or five of these questions on each test.

Below you’ll find a link to 10 sample system of equations questions (and a second link to the solutions).

But first, let’s do a little review.

Most systems of linear equations have a single ordered pair as their solution. It’s the point where the two lines intersect, and you can find the coordinates of that point by graphing the lines on  your calculator or using the substitution or elimination methods. But what about the special cases? Let’s take a look at systems that have either an infinite number of solutions or no solution at all.

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SAT Math: A Challenging Functions Problem

As my students and I have prepared for the new SAT, one of the things we’ve noticed is that the more challenging Math problems aren’t just computational or problems where you can throw something into your calculator to get the answer. They are more theoretical — you have to really know the math in order to come up with the correct answer.

Let’s consider a challenging problem involving functions.

A function is defined by the equation below.

f(x) = a(x – 2)(x – 10)

If the minimum value of the function is -8, what is the value of a?

A)  -2
B)  1/4
C)  1/2
D)  2

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