sat math Archives - Page 6 of 24

500 New SAT Math Problems – Just 19.99 Today Only

June 30, 2024

$500 New SAT Math Problems$

500 New SAT Math Problems
Just 19.99 on Amazon

Hi everyone! The latest edition of 500 New SAT Math Problems is now available in paperback from Amazon. This edition just has been modified from the previous edition to account for the changes on the Digital SAT.

~~The paperback is now on sale on Amazon for only $19.99. Note that once the sale ends (by the end of today), the price of this book will go up to $42.99.~~

The promotion has ended. Thanks to everyone who participated. The book is now available at its regular price here: 500 New SAT Math Problems

If you have any questions, feel free to contact me at steve@SATPrepGet800.com

Thank you all for your continued support!

A Trick For Free Two Day Shipping

I would like to finish this post with a little trick you can use to get free 2 day shipping on any of the books you decide to purchase without making any additional purchases. If you have never used Amazon Prime you can sign up for a free month using the following link.

If you have already had a free trial of Amazon Prime you can simply open up a new Amazon account to get a new free trial. It just takes a few minutes! You will need to use a different email address than the one you usually use.

This next part is very important! After you finish your transaction, go to your Account, select “Manage my prime membership,” and turn off the recurring billing. This way in a month’s time Amazon will not start charging you for the service.

After shutting off the recurring billing you will still continue to receive the benefit of free 2 day shipping for one month. This means that as long as you use this new Amazon account for your purchases you can do all of your shopping on Amazon for the next month without having to worry about placing minimum orders to get free shipping.

Just be aware that certain products from outside sellers do not always qualify for free shipping, so please always check over your bill carefully before you check out.

Well I hope you decide to take advantage of this very special offer, or at the very least I hope you will benefit from my Amazon “free 2 day shipping trick.” Here is the link one more time:

If you think your friends might be interested in this special offer, please share it with them on Facebook:

Thank you all for your continued support!

Derivation of the Quadratic Formula

June 18, 2017

Derivation of the Quadratic Formula

Last week we went over how to solve quadratic equations using the quadratic formula. You can see those posts here: Quadratic Formula 1 2

Recall that the solutions to the quadratic equation ax² + bx + c = 0 are

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

At the end of the second post, I posed the following problem:

Challenge Problem: Solve the general quadratic equation ax² + bx + c = 0 by completing the square, and note that this gives a derivation of the quadratic formula.

I will now provide a detailed solution. You may want to review the following posts before reading the following solution:

Square Root Property Completing the Square Solving Quadratic Equations

Solution:

$ax^2+bx+c=0$

$ax^2+bx=-c$

$x^2+\frac{b}{a} x=-\frac{c}{a}$

$x^2+\frac{b}{a} x+(\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2$

$(x+\frac{b}{2a})^2=-\frac{c}{a}+\frac{b^2}{4a^2}$

$(x+\frac{b}{2a})^2=\frac{b^2}{4a^2}-\frac{c}{a} (\frac{4a}{4a})$

$(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}$

$x+\frac{b}{2a}=\pm \frac{\sqrt{b^2-4ac}}{2a}$

$x=-\frac{b}{2a}\pm \frac{\sqrt{b^2-4ac}}{2a}$

$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

The derivation above was challenging. Try to understand each step. If you have any question on a particular step, feel free to post your question in the comments.

If you liked this article, please share it with your Facebook friends:

The Quadratic Formula – Part 2

June 16, 2017

The Quadratic Formula
Part 2

Yesterday we began discussing the quadratic formula, and I asked you to solve a specific quadratic equation using the formula. You can see that post here: Solving Quadratic Equations with the Quadratic Formula

Today I will provide the solution to that problem. For your reference, let me give you the quadratic formula one more time:

The solutions to the quadratic equation ax² + bx + c = 0 are

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Example: Solve the quadratic equation x² – 2x – 15 = 0 by using the quadratic formula.

Solution:

$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$=\frac{2\pm \sqrt{(-2)^2-4(1)(-15)}}{2\cdot 1}$

$=\frac{2\pm \sqrt{4+60}}{2}$

$=\frac{2\pm \sqrt{64}}{2}$

$=\frac{2\pm 8}{2}$

$=\frac{2}{2}\pm \frac{8}{2}$

$=1\pm 4$

So, the two solutions are 1 + 4 = 5, and 1 – 4 = –3

Note: This particular problem could be solved more easily by factoring.

Why does the quadratic formula look so messy? It’s not a very pleasant looking formula. It would have been nicer if we had a simpler formula for solving a quadratic equation. But unfortunately, this is simply what it turned out to be. As a challenging exercise, I would like you to see this for yourself firsthand. I will give a full solution tomorrow:

Challenge Problem: Solve the general quadratic equation ax² + bx + c = 0 by completing the square, and note that this gives a derivation of the quadratic formula.

If you liked this article, please share it with your Facebook friends:

Solving Quadratic Equations with the Quadratic Formula

June 15, 2017

Solving Quadratic Equations
with the Quadratic Formula

A quadratic equation has the form ax² + bx + c = 0.

The quadratic formula is a formula that gives us all solutions to a quadratic equation right away. The formula is as follows:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Example: Solve the quadratic equation x² – 2x – 15 = 0 by using the quadratic formula.

We have previously solved this problem by completing the square. You can see that solution here: Solution by Completing the Square

Try to solve this problem by using the quadratic formula. I will give the full solution tomorrow.

If you liked this article, please share it with your Facebook friends:

Hard Functions Problem with Solutions for the SAT

April 22, 2017

Hard Functions Problem

with Solutions for the SAT

Today I will give several solutions to the problem I posted yesterday. Use this link to see the question: Hard Function Problem for the SAT

This problem is from the book 28 SAT Math Lessons – Advanced Course. Here is the problem one more time, followed by three methods of solution.

Level 5 Functions

For which of the following functions is it true that f(-x) = f(x) for all values of x ?

A) f(x) = x² + 5
B) f(x) = x² + 5x
C) f(x) = x³ + 5x
D) f(x) = x³ + 5

Solution by picking numbers: Let’s choose a value for x, say x = 2. We compute f(-2) and f(2) for each answer choice.

f(-2) f(2)
A) 9 9
B) -6 14
C) -18 18
D) -3 13

Since choices (B), (C), and (D) do not match up, we can eliminate them. The answer is therefore choice (A).

Important note: (A) is not the correct answer simply because both computations gave the same answer. It is correct because all 3 of the other choices did not work. You absolutely must check all four choices!

* Quick solution: We are looking for an even function. Each answer choice is a polynomial. Therefore, the answer is the one with only even powers of x. This is choice (A) (remember that 5 = 5x⁰ ).

Note: See the following post for more information on this method: Even and Odd Functions

Graphical solution: Begin putting each of the four answer choices into your graphing calculator (starting with choice (B) or (C)), and choose the one that is symmetrical with respect to the y-axis. This is choice (A).

Note: Again, see this post for more information: Even and Odd Functions

More SAT and ACT Math Problems with Explanations

If you are preparing for the SAT, ACT, or an SAT math subject test, you may want to take a look at the Get 800 collection of test prep books.

And if you liked this article, please share it with your Facebook friends:

Hard Functions Problem for the SAT

April 21, 2017

Hard Functions Problem for the SAT

Today I would like to give a hard functions problem from the book 28 SAT Math Lessons – Advanced Course. I will provide a full explanation for this problem tomorrow. Feel free to leave your own solutions in the comments.

Level 5 Functions

For which of the following functions is it true that f(-x) = f(x) for all values of x ?

A) f(x) = x² + 5
B) f(x) = x² + 5x
C) f(x) = x³ + 5x
D) f(x) = x³ + 5

If you’re having trouble getting started, you may want to read the following post: Even and Odd Functions

The Differences Between the SAT Math Subject Tests

In addition to the SAT reasoning test (which most people simply refer to as the SAT), the College Board offers several SAT subject tests. Whether you should take any of these tests depends entirely upon the colleges you will be applying to. Some colleges require 2 or 3 SAT subject tests, while others may just recommend them.

There are two different subject tests for math: a Level 1 test and a Level 2 test. The question that I get asked about most frequently with regard to these tests is which is the best to take. In this article, I hope to provide you with enough information to make an informed decision.

Test Length and Structure

Both the Level 1 and Level 2 SAT math subject tests consist of 50 multiple choice questions. You are given an hour to complete each test. I like to think of each test as broken up into 5 pieces of 10 questions each, progressively increasing in difficulty level. In other words, I consider questions 1 through 10 to be difficulty level 1, questions 11 through 20 difficulty level 2, and so on, up through difficulty level 5 questions. Note that this is just a rough estimate – overall the questions do increase in difficulty as you work your way through the test, but easier questions can certainly occur after harder questions. In an attempt to avoid confusion, from now on I will use the word “Level” when talking about the Level 1 or 2 test, and the abbreviations L1, L2, L3, L4, and L5 for the difficulty levels of questions.

One really important thing to understand is that an L1 question on the Level 2 test is much harder than an L1 question on the Level 1 test. As an estimate, I like to add 2 “L’s” in translating from the Level 2 to the Level 1 test. In other words, an L1 question on the Level 2 test (questions 1-10) is about the same difficulty as an L3 question on the Level 1 test (questions 21-30). As another way to think of this, to translate the difficulty level of a problem from the Level 1 to the Level 2 test, simply subtract 20 from the question number. So for example, question 40 on the Level 1 test would probably appear as about question 20 on the Level 2 test, and question 15 on the Level 1 test wouldn’t appear on the Level 2 test because it would be too easy.

Scoring

Both SAT math subject tests are graded out of 800, but the way the score is determined is quite different for these two tests. The Level 2 test has a much, much more generous curve than the Level 1 test. If you get 2/3 of the answers correct on the Level 1 test, your final score will be about a 600. The same 2/3 ratio of correct answers on the Level 2 test will give you a final score of about 700. On many Level 2 tests, you can get up to 6 questions wrong and still get a perfect score of 800, whereas on the Level 1 test, you usually need to get every question correct for a perfect score.

Mathematics Covered

The two SAT math subject tests cover many of the same topics, but there are differences. The Level 1 test covers topics from two years of algebra and one year of geometry (including a bit of trigonometry). The Level 2 test covers more advanced trigonometry and precalculus, and even throws in the occasional calculus problem. The following topics are covered on the Level 2 test, but not on the Level 1 test.

Series
Vectors
Graphical Analysis of Complex Numbers
Quadratic and Exponential Regression
Standard Deviation
Parametric Equations
Recursive Equations
Ellipses and Hyperbolas
Polar Coordinates
Distance in Three Dimensions
Solid Geometry
Radian Measure
Graphs of Trig Functions
Inverse Trig Functions
Laws of Sines and Cosines
Double Angle Formulas

One other big difference is that the Geometry on the Level 1 test is mostly coordinate and plane geometry, whereas there are only coordinate and solid geometry questions on the Level 2 test (although the solid geometry questions very often amount to understanding concepts in plane geometry).

Final Note

The best way to decide which test you should focus on is to take a practice test for both the Level 1 and Level 2 subject tests. I strongly recommend using official practice tests from the test makers themselves. Make sure to finish each test, and to score each one carefully. Remember that the Level 2 test has a much more generous curve. So it is possible to “feel” like you are performing much worse on the Level 2 test, and yet still wind up with a higher scaled score.

Once you decide which test you will focus on, make sure you take some time to prepare. If you’ve already prepared for SAT math, then about 20 minutes per day for another month should do the trick. You may want to use one of the books from my “320 Problems” series to prepare. Click on each of the following books to view the book’s Amazon page:

$320-SAT-Math-Subject-Test-1-Front-Cover$

$320-sat-math-subject-test-problems-front-cover$

If you liked this article, please share it with your Facebook friends:

SAT Verbal Prep Book – New Edition

January 12, 2017

SAT Verbal Prep Book – New Edition

Today I would like to announce the release of a new edition of the SAT Verbal Prep Book for Reading and Writing Mastery.

The paperback is now on sale on Amazon. Click the following link to get to the book’s Amazon page: SAT Verbal Prep Book

Note that this new edition has many significant changes from its predecessor. Theses changes include:

additional strategies for better performance on the reading, language and essay sections of the SAT.
4 reading tests and 4 language tests in the format of the actual SAT.
more questions requiring the analysis of charts, graphs, and other figures.
diagrams demonstrating the proper way to mark up passages to make it easy to find the most important information.

Take a look at my product page to see all of my books: Get 800 Product Page

A Trick For Free Two Day Shipping

Just be aware that certain products from outside sellers do not always qualify for free shipping, so please always check over your bill carefully before you check out.

Well I hope you decide to take advantage of this very special offer, or at the very least I hope you will benefit from my Amazon “free 2 day shipping trick.” Here is the link one more time:

If you think your friends might be interested in this special offer, please share it with them on Facebook:

Thank you all for your continued support!

Hard Statistics Problem with Solution

January 4, 2017

$320 SAT Math Subject Test Problems - Level 2$

Hard Statistics Problem with Solution

Today I would like to give a solution to yesterday’s statistics problem. Before reading the solution, you may want to review the following article: Changing Averages to Sums

Here is the problem once again, followed by a solution.

Level 5 Statistics

Suppose that the average (arithmetic mean) of a, b, and c is h, the average of b, c, and d is j, the average of c, d, and e is k, and the average of d, e, and f is m. What is the value of (a + c) – (d + f ) ?

A) h – j + k – m
B) 3(h – j + k – m)
C) 3/4 (h – j + k – m)
D) 3(h + j + k + m)
E) 3/4 (h + j + k + m)

Solution by changing averages to sums: We have

a + b + c = 3h

b + c + d = 3j

c + d + e = 3k

d + e + f = 3m

We subtract the second equation from the first equation to get

a – d = 3h – 3j

We subtract the fourth equation from the third equation to get

c – f = 3k – 3m

We now add these last two equations to get

a – d + c – f = 3h – 3j + 3k – 3m

This is equivalent to choice B.

Hard Statistics Problem

Today I would like to give a hard statistics problem from the book 320 SAT Math Subject Test Problems – Level 2. I will provide a full explanation for this problem tomorrow. Feel free to leave your own solutions in the comments.

Level 5 Statistics

A) h – j + k – m
B) 3(h – j + k – m)
C) 3/4 (h – j + k – m)
D) 3(h + j + k + m)
E) 3/4 (h + j + k + m)

Is Your SAT Math Score Between 500 and 600?

“28 SAT Math Lessons to Improve Your Score in One Month – Intermediate Course – For Students Currently Scoring Between 500 and 600 in SAT Math.”

The Old Schoolhouse (a website for homeschoolers) just posted a nice comprehensive review of this book here:

http://www.theoldschoolhouse.com/product-reviews/?rid=5942

Click the image at the top of this page to view the book on Amazon.

Newer posts

Older posts

500 New SAT Math Problems Just 19.99 on Amazon

A Trick For Free Two Day Shipping

Derivation of the Quadratic Formula

The Quadratic Formula Part 2

Solving Quadratic Equations with the Quadratic Formula

Hard Functions Problem

with Solutions for the SAT

Level 5 Functions

Hard Functions Problem for the SAT

Level 5 Functions

More SAT and ACT Math Problems with Explanations

The Differences Between the SAT Math Subject Tests

Test Length and Structure

Scoring

Mathematics Covered

Final Note

SAT Verbal Prep Book – New Edition

Hard Statistics Problem with Solution

Level 5 Statistics

More SAT and ACT Math Problems with Explanations

Hard Statistics Problem

Level 5 Statistics

More SAT and ACT Math Problems with Explanations

Is Your SAT Math Score Between 500 and 600?

500 New SAT Math Problems
Just 19.99 on Amazon

The Quadratic Formula
Part 2

Solving Quadratic Equations
with the Quadratic Formula