I’ve said it before and I’ll say it again: SAT math can be a bear! That said, there are certain math formulas, identities, and processes that you can learn that will help you efficiently and effectively navigate SAT math. So, this article will discuss 10 high-value SAT math formulas and facts you should know for the test.
Here are the topics we’ll cover:
- What Is Tested in SAT Math
- How and When Are SAT Math Problems Presented on the SAT?
- The SAT Reference Sheet
- #1: The Quadratic Formula
- #2: The Discriminant
- #3: The Percent Change Formula
- #4: The Pythagorean Theorem
- #5: Multiplying Exponents with the Same Base
- #6: The Median of a Large Data Set
- #7: Slope-Intercept Form of a Line
- #8: Slope Formula
- #9: Difference of Squares
- #10: SOHCAHTOA
- Summary
- What’s Next?
Before we work on the 10 items, let’s take a quick look at what is tested in SAT math.
What Is Tested in SAT Math
If you are just starting out with your SAT prep, you may initially feel intimidated by SAT math because the folks who have created the SAT (the College Board) have broken down SAT Math into rather broad categories:
- Heart of Algebra
- Problem Solving and Data Analysis
- Passport to Advanced Math
- Additional Topics in Math
The topics above may not seem to relate to anything you have done thus far in school, but don’t worry! Those categories include topics that are quite relatable to much of what you’ve already seen in your high school math classes. So, let me show you what major topics are actually tested on the SAT.
KEY FACT:
The College Board has divided SAT math into four major categories.
The TTP Breakdown of What is Tested in SAT Math
Here are the major math topics you need to learn for SAT math.
- Solving Linear Equations
- Inequalities and Absolute Value
- Coordinate Geometry
- Linear Functions
- Systems of Equations
- General Word Problems
- Rates
- Unit Conversions
- Proportions and Ratios
- Statistics
- Percentages
- Exponents
- Radicals
- Quadratic Equations and Polynomials
- Functions
- Complex Numbers
- Geometry
- Trigonometry
- Probability
I bet many of the topics above seem very familiar! Furthermore, all of the above topics can be learned. In fact, we wrote an awesome article about how to prepare for SAT math. So, you don’t need to stress too much.
TTP PRO TIP:
There are 19 major SAT math topics.
Before jumping into our top SAT math formulas, let’s discuss how math shows up on the SAT.
How and When Are SAT Math Problems Presented on the SAT?
The SAT has two math sections. The No Calculator section has 15 multiple-choice questions and 5 grid-ins. You have 25 minutes to complete it. The Calculator section has 30 multiple-choice and 8 grid-in questions, and you have 55 minutes to complete it.
Now, you may ask yourself, how can I complete so many questions in so little time? Well, the good news is that if you memorize common formulas and equations, you can attack math questions quickly and accurately! Yes, it will take time and effort to memorize the necessary formulas, but in the end, it will all be worth it!
TTP PRO TIP:
Memorizing many SAT math equations will help you quickly attack SAT math problems.
Before we get into which formulas to memorize, one last thing to discuss is the SAT formula sheet.
The SAT Reference Sheet
You will have an SAT math formula sheet given to you to use as a reference during your exam:
So, the question now is, how should you use this sheet? As you can see, the sheet is actually quite limited. For example, it does not contain common geometry formulas such as the circumference of a circle or coordinate geometry formulas such as the distance formula.
It does have a few geometry formulas that you might find useful, but since it’s located in front of your SAT booklet, every time you refer to it, you are wasting valuable time. So, it’s great to use for backup if you’ve forgotten something. However, in an ideal world, you will have memorized the formulas and facts on the sheet during your SAT prep.
TTP PRO TIP:
Even though the SAT provides some formulas in the front of your booklet, to save time, you should have them memorized.
Now, let’s discuss some of the top formulas to memorize and understand for your SAT.
#1: The Quadratic Formula
Whether you learned it by poem, by song, or by rote memorization, the quadratic formula is a mainstay of high school algebra! If you can’t easily factor a quadratic, you can use the quadratic formula to determine the roots of a quadratic equation, which are also the x-coordinates (if they exist) of the points where the parabola crosses the x-axis.
To use the quadratic formula, you need to express the quadratic equation as ax^2 + bx + c = 0. From there, use care to substitute the values of a, b, and c into the quadratic formula.
TTP PRO TIP:
When you can’t easily factor a quadratic in the form of ax^2 + bx + c = 0, use the quadratic formula to determine the roots of the equation.
Let’s practice with an example.
Quadratic Formula Example
What is the sum of the roots of the equation 3x^2 – 7x + 2?
- 1/3
- 4/3
- 5/3
- 7/3
Solution:
This quadratic equation is difficult to factor by traditional methods of factoring. In a case such as this one, using the quadratic formula is an effective means of determining the roots of the equation.
First, we identify that a = 3, b = -7, and c = 2. We substitute these values into the quadratic formula:
The two roots of the quadratic equation are 2 and 1/3, so their sum is 2 + 1/3 = 6/3 + 1/3 = 7/3.
Answer: D
Next, let’s see how we can use discriminant analysis to determine the number and types of real solutions to a quadratic equation.
#2: The Discriminant
The discriminant is the expression inside the square root symbol of the quadratic formula. Instead of using the entire quadratic formula, we can use the discriminant to answer questions about the number and types of roots of a quadratic equation.
The discriminant is:
b² – 4ac
To use the discriminant, we analyze any quadratic equation that is expressed in the following form:
ax² + bx + c = 0
After we evaluate the discriminant, we can say the following:
- If the discriminant is positive, then we have 2 real roots.
- If the discriminant is zero, then we have 1 real root (repeated root).
- If the discriminant is negative, then the quadratic produces no real roots.
TTP PRO TIP:
Evaluating the discriminant will tell us the number and type of roots of a quadratic equation.
So, now that we have discussed the discriminant, you may be wondering, how will this be tested on the SAT? Let’s practice an example now!
Discriminant Example
3x² = 12 – 2x has how many real roots?
- 0
- 1
- 2
- more than 2
Solution:
To solve, let’s put the equation in standard form, so we can use our discriminant formula.
3x² + 2x – 12 = 0
We see that a = 3, b = 2, and c = -12. Substituting these values into the formula for the discriminant, we have:
b² – 4ac:
(2)² – 4(3)(-12)
4 – 12( -12)
4 + 144 = 148
Since the result is positive, we know that there are 2 real roots.
Answer: C
Next, let’s discuss the percent change formula.
#3: The Percent Change Formula
When dealing with the math topic of percents, you may have encountered questions about percent change, percent decrease, and percent increase. All of these can be answered using the percent change formula, which is the following:
Percent change = [(new value – old value) / (old value)] x 100%
The only caveat to consider is that, when solving for the percent decrease, you must end up with a negative result, and when solving for the percent increase, you must end up with a positive result. In fact, to show how this formula works, we can practice a problem on percent increase and percent decrease.
TTP PRO TIP:
The percent change formula is percent change = [(new value – old value) / (old value)] x 100%. A percent increase occurs when the result is positive, and a percent decrease occurs when the result is negative.
Percent Increase Example
In 2018, the price of a house was valued at $800,000. In 2021, the price of the same house was valued at $960,000. What was the percent increase in the house’s value between 2018 and 2021?
- 10%
- 20%
- 25%
- 30%
Solution:
To determine the percent increase, we will use the following formula:
Percent change = [(new value – old value) / (old value)] x 100%
Since we are asked for a percent increase, the result must be positive. Thus, we can let the new value = 960,000 and the old value = 800,000. Plugging these values into the formula, we have:
Percent change = [(new value – old value) / (old value)] x 100%
Percent change = [(960,000 – 800,000) / (800,000)] x 100%
Percent change = (160,000 / 800,000) x 100%
Percent change = 0.2 x 100% = 20%
Answer: B
Next, let’s practice a similar problem, but this time we’ll calculate a percent decrease.
Percent Decrease Example
Last year, the cost of gravel was 65 dollars per yard. If, this year, the cost of gravel is 40 dollars per year, what was the approximate percent decrease in the cost of gravel between last year and this year?
- 9%
- 16%
- 20%
- 23%
Solution:
To determine the percent decrease, we will use the following formula:
Percent change = [(new value – old value) / (old value)] x 100%
Since we are asked for a percent decrease, the result must be negative. Thus, we can let the new value = 40 and the old value = 65. Plugging these values into the formula, we have:
Percent change = [(new value – old value) / (old value)] x 100%
Percent change = [(40 – 65) / ( 65)] x 100%
Percent change = [(-15) / (65)] x 100% = -0.23 x 100% = -23%
Thus, the percent decrease was 23%.
Answer: D
Next, let’s discuss an important geometry formula, the Pythagorean theorem.
#4: The Pythagorean Theorem
There are many formulas and equations worth memorizing in geometry, and the Pythagorean theorem is certainly a major one.
The Pythagorean theorem states that in any right triangle, A2 + B2 = C2, where C is the length of the triangle’s hypotenuse, and A and B are the lengths of the triangle’s legs. This can be more clearly shown in the diagram below:
So, for example, if we know that two legs of a right triangle have lengths of 3 and 4, we can determine the length of the hypotenuse as follows:
A2 + B2 = C2
32 + 42 = C2
9 + 16 = C2
25 = C2
Thus, C must equal 5.
TTP PRO TIP:
We should memorize the Pythagorean theorem: A2 + B2 = C2.
Let’s now practice with an example.
Pythagorean Theorem Example
If the length of the hypotenuse of a certain right triangle is 10 and one of the legs is equal to 8, what is the area of the triangle?
- 10
- 24
- 28
- 36
Solution:
Since we have two sides of a right triangle and need the third side, we can use the Pythagorean theorem.
A2 + 82 = 102
A2 + 64 = 100
A2 = 36
A = 6
(Note: we could have recognized that the right triangle above was a 6-8-10 right triangle, which is known as a special right triangle.)
Now, we see that the lengths of the two legs are equal to 6 and 8, so we can use those lengths to determine the area. The area of a triangle formula is as follows:
Area = 1/2 * base * height
Area = 1/2 * 6 x 8
Area = 1/2 * 48
Area = 24
Answer: B
Next, let’s discuss a technique that often trips up students: multiplying exponents with the same base.
#5: Multiplying Exponents with the Same Base
There are a variety of rules to memorize when it comes to exponents, but one commonly used rule is the following:
(x^a)(x^b) = x^(a+b)
What this rule says is that when you multiply two exponents with the same base, you add the exponents. For example:
- 2^4 * 2^7 = 2^(4 + 7) = 2^11
- a^6 * a^14 = a^ (6 + 14) = a^20
TTP PRO TIP:
Memorize the following rule for multiplying exponents: x^a * x^b = x^(a+b).
Let’s practice with an example.
Multiplying Exponents with the Same Base Example
If 2x + 2y = 12 and (2x)(2y) = 2a, what is the value of a?
- 6
- 8
- 10
- 12
Solution:
To solve this question, we will use the following rule:
(x^a)(x^b) = x^(a+b)
Thus, since (2x)(2y)=2a, we have:
2x+y = 2a
Thus, it’s clear that x + y = a.
Since we also know that 2x + 2y = 12, we can divide that equation by 2, obtaining x + y = 6.
Thus, 2^a = 2^(x + y) = 2^6, and so a = 6.
Answer: A
Next, let’s discuss an important formula used frequently in statistics that we use to determine the median of a large data set.
#6: The Median of a Large Data Set
You may recall that the median of a set of data values is found by first listing the values in ascending or descending order, and then determining the middle value. For example, say we have the following data set:
2, 7, 10, 13, 44
We know that the median is 10, the middle value. Note that in this example, we have an odd number of values, so the median is an actual data value. But if the number of items in the data set is even, we must calculate the average of the two middle values. Consider this data set:
4, 7, 11, 15, 19, 22, 30, 45
Here, there is no single middle value. The median, then, is the average of the two middle values. So, in this example, the median is (15 + 19) / 2 = 17.
When we have a large data set containing n data values, the methods described above will be unwieldy. Instead, we use a formula to determine the position of the median in the data set:
Position of median = (n + 1) / 2
TTP PRO TIP:
When values of a large data set are listed in ascending order, we can determine the position of the median by using the formula (n + 1) / 2 , where n = the number of data values in the set.
Let’s use this formula with an example.
Median of a Large Data Set Example
The finishing times, in seconds, of a race of 21 third-grade students are listed below. What is the median finishing time?
14, 14, 15, 15, 16, 16, 16, 17, 17, 19, 20, 22, 23, 23, 25, 26, 27, 28, 29, 29, 42
- 18
- 20
- 22
- 24
Solution:
First, notice that the finishing times of the n = 21 students are already listed in ascending order. Thus, the position of the median is:
Position of median = (n + 1) / 2 = (21 + 1) / 2 = 22 / 2 = 11
The 11th data value is the median. We count until we get to the 11th data value, and we see that the median finishing time is 20 seconds.
Answer: B
Let’s now take a look at the slope-intercept form of a line.
#7: Slope-Intercept Form of a Line
In high school algebra class, even more pervasive than the quadratic formula is the slope-intercept form of a line. If you haven’t memorized it yet, do so before you take the SAT. Let’s take a look at what it is and how it’s used.
The point-slope form of a line is expressed as:
y = mx + b
y = the y-coordinate of a point on the line
m = the slope of the line
x = the x-coordinate of a point on the line
b = the y-intercept of the line
On the SAT, you might be asked questions about any of the quantities in the slope-intercept form of a given line.
TTP PRO TIP:
The slope-intercept form of a line is y = mx + b, where x and y are the coordinates of a point on the line, m is the slope of the line, and b is the y-intercept of the line.
For example, if you are given the equation y = 5x – 3, you might be asked to identify the slope (m = 5), the y-intercept of the line (-3), or the coordinates of a particular ordered pair located on the line. Let’s look at this in more detail, with an example.
Slope-Intercept Form of a Line Example
Point (w, 4) lies on line q. If the equation of line q is y = (3/4)x + 8, what is the value of w?
- -16/3
- -8/3
- -⅓
- ⅓
Solution:
Let’s plug the y-coordinate of the point (w, 4) into the equation and solve for x.
4 = (3/4)x + 8
-4 = (3/4)x
-16/3 = x
The x-coordinate of the point (w, 4) is -16/3. Thus, the value of w is -16/3.
Answer: A
Often, you will not be provided the value of the slope of a line. You will have to calculate it by using the slope formula.
#8: Slope Formula
The calculation of the slope of a line passing through the points (x1, y1) and (x2, y2) is made using the following formula:
where y1 and y2 are the y-coordinates of the two ordered pairs and X1 and x2 are the x-coordinates of the two ordered pairs.
TTP PRO TIP:
The slope of a line passing through any two points on the line, (x1, y1) and (x2, y2), can be determined by using the formula.
Let’s try an example.
Slope Formula Example
Line p passes through the points (5, -2) and (1, 3). What is the slope of line p?
- -5/4
- -1/4
- 1/6
- 5/6
Solution:
We substitute the x- and y-values of the two ordered pairs into the slope formula:
Answer: A
Let’s turn our attention to another important algebra topic often encountered on the SAT: the difference of squares.
#9: Difference of Squares
The difference of squares is an algebraic identity that appears frequently on the SAT and other standardized exams. Become familiar with it and how it is used, in order to have another tool in your toolkit!
The difference of squares identity is a² – b² = (a – b)(a + b).
TTP PRO TIP:
The difference of squares identity is a² – b² = (a – b)(a + b).
A simple example of this is to recognize that the expression x² – 9 can be put into the factored form of (x – 3)((x + 3). However, SAT math questions will not be so direct. Let’s look at a more representative example.
Difference of Squares Example
If (9s² – 81) = 0, then 5s – 7 could be equal to which of the following?
- -52
- -22
- 0
- 38
Solution:
First, we factor out the common factor of 9 from the expression (9s^2 – 81), and then we factor using the difference of squares identity:
9(s² – 9) = 0
9(s – 3)(s + 3) = 0
s – 3 = 0 OR s + 3 = 0
s = 3 s = -3
We have two possible values for s, so we plug those values into the expression 5s – 7:
For s = 3, the value of 5s – 7 = (5)(3) – 7 = 15 – 7 = 8. This is not one of the answer choices, so we next consider the second possible value of s.
For s = -3, the value of 5s – 7 = (5)(-3) – 7 = -15 – 7 = -22. This is one of the answer choices.
Answer: B
Our final SAT concept comes from trigonometry.
#10: SOHCAHTOA
No list of important equations and identities would be complete without mention of SOHCAHTOA, an acronym that helps us remember the three basic trigonometric functions of sine, cosine, and tangent.
If we have a right triangle with angle θ, we identify the side opposite the right angle as the hypotenuse, the side of the triangle that “touches” angle θ as the adjacent, and the side that doesn’t touch angle θ as the opposite.
We define the following:
sin θ = Opposite / Hypotenuse ⇒ SOH
cos θ = Adjacent / Hypotenuse ⇒ CAH
tan θ = Opposite / Adjacent ⇒ TOA
On the SAT, you won’t encounter highly complicated questions involving SOHCAHTOA. So, let’s look at a basic example.
SOHCAHTOA Example
For the right triangle given above, what is the value of tan M?
- 2/7
- 2√5 / 15
- 3√5 / 2
- 5/2
Solution:
Using SOHCAHTOA, we know that tan M = Opposite / Adjacent. In order to determine the value of the side opposite angle M, we use the Pythagorean theorem:
A2 + B2 = C2
22 + B2 = 72
4 + B2 = 49
B2 = 45
B = 3√5
We now know that the side opposite angle M is 3√5. From the figure, we know that the side adjacent to angle M is 2, so we can use SOHCAHTOA to determine the tangent of angle M:
TOA ⇒ tan M = Opposite / Adjacent
tan M = 3√5 / 2
Answer: C
Summary
There are four major categories that the College Board has identified for testing on the math section of the SAT:
- Heart of Algebra
- Problem Solving and Data Analysis
- Passport to Advanced Math
- Additional Topics in Math
Included in these four categories are 19 major math topics, ranging from linear equations to percents to statistics. It’s critical to study these topics and to master the necessary concepts, so that no matter what is asked on test day, you can confidently and accurately answer every question.
In this article, we have presented you with 10 formulas and identities which will help immeasurably with your performance on the SAT math section. They include:
- The Quadratic Formula
- The Discriminant
- The Percent Change Formula
- The Pythagorean Theorem
- Multiplying Exponents with the Same Base
- The Median of a Large Data Set
- Slope-Intercept Form of a Line
- Slope Formula
- Difference of Squares
- SOHCAHTOA
There are hundreds of formulas and concepts that you need to know for the SAT. Memorizing these 10 and knowing how to use them is a great foundation for earning a high SAT score! So, make an SAT formula sheet, and get these must-know formulas memorized!
What’s Next?
Doing well on the SAT math section is only part of the equation for getting a great SAT score. You also need to have a solid strategy for studying for the SAT. These 5 steps for getting the right start on your SAT preparation will help.
Good luck!
