[Music] what's up everybody and welcome back to another video from the scaler Learning Channel on the digital SAT Math this is a video that is so important so critical it's going to be so helpful for everybody studying and preparing for the digital sat and this is the digital sat math formula Bible it's got all the critical Concepts and formulas that you need to be successful on the digital sat and by the way if you guys are looking for the best resource on planet Earth to study for the digital sat you got to check out the SAT crash course the link is in the description below and if you use the code scaler you'll get 20% off not only do they have 10 practice digital SATs they also have my video solutions to each and every math problem on those practice tests so you can make sure that you understand everything when you w walk in on test day without further Ado let's do it all right so starting off with item zero right everything's numbered and this is item zero and there's a reason for that because this is actually just referencing all of the given formulas this is the reference sheet that is accessible during and throughout both math portions module one and module two of the digital sat you can pull this up anytime and all these formulas here it's got area and circumference of a circle it's got conversions between radians and degrees right it's showing you right here that 360 degrees is equal to 200 2 pi radians and it's telling you that the sum of angles in a triangle is 180 degrees all these great volume formulas and even special right triangles so a lot of these formulas I'm not including in this digital sat math formula Bible with the exception of special right triangles and Pythagorean theorem just because they're so important so I'm going to reiterate those but the rest of these are not there for a reason because you can find them right here so get familiar with these formulas get familiar with how to pull them up so you can use them in the best way possible so formula number one is how to calculate slope of a line when given to coordinates so let's check it out right it's m equal Y 2 - y1 over X2 - X1 and here's an example right let's say we got coordinates 7251 and by the way how you label X2 and x and Y2 X1 and y1 is arbitrary just make sure you stay consistent with the subtraction now we're going to subtract and in this format we're going to go top minus bottom right so we've got 2 - one on the top the difference of the Y's over 7 - 5 on the bottom the difference of the x's and that leaves us with 1/ 2 and that's your final answer for the slope boom done formula number number two is slope intercept form for a linear equation this is super important and that is y = mx plus b where m is the slope of a line B is the Y intercept so just so you can see a quick example we got y = 2x + 7 we got a slope of Two And A Y intercept of seven now if you looked at this on a graph it would look like this there's our nice Y intercept right at seven that's where it hits the Y AIS and the slope is two which can also be Rewritten as 2 over one so that's a rise of Two and a run of one 1 meaning a horizontal distance of one so that's how you do it next we got point slope form this is also super important y - y1 equals m which is the slope * x - X1 so again m is our slope and X1 and y1 what is that those are just values it's one point any point on a line you can plug that in and you'll get a nice equation of the line so here we have an example we have y - 4 = 3 * x - 3 so the slope of course is 3 and the coordinate would be 3A four so if we look at the graph here you can see there's 34 right there on the graph and then we got that rise of three and a run of two that's how you do it done formula number four is how to find the midpoint of two points okay so it what you basically doing is you're taking the average of the X values of the two points the average of the yv values of the two points and you got it and that makes a lot of sense because midpoint is in the middle you find that by taking the average so for example right if we've got this as our formula if we we take the points 8 4 6 7 I'm going to take the average of 8 and six I'm going to take the average of 4 and 7 so that's going to be 14 over2 which is 7 11 over 2 11 Hales or 5.5 next we get to the distance formula so this isn't used a ton necessarily but a lot of times it's really helpful especially if you're trying to calculate the radius of a circle and you're given the center of the circle and a point on the exterior of the circle so here we go here's the distance formula distance equals the difference of the x value squ plus the difference of the y- Val squared and you know the square root of that so let's check it out here if we have two points for example right 53 and 1 0 if we want to find the distance between these two plug and chug right 5 - 1 2 + 3 - 0 2 5 - 1 is 4 4 2 is 16 + 3 - 0 that's 3 squar is 9 so we got 16 + 9 is 25 and the square root of 25 is five so that's how you do it done next we got length of an arc okay and what's an arc we're talking about on a circle it's like the little crust it's a piece of the circumference hence the formula right this makes a lot of sense it's n over 360 n is the central angle that cuts out this Arc times 2 pi r now you'll remember 2 pi r is the circumference of a circle so we're taking a piece of that circumference right so if we look at this graph it makes a lot of sense again it's that crust of the pizza and is that central angle R is the radius that's how you do it next we're going to come to area of a sector which is very similar to the length of an Arc right we have a similar formula and now we're taking the central angle n over 360 and instead of multiplying it by 2i R which is a circumference we multiply it by the area of a circle which is pi r SAR so if we look at the graph now the sector it's like the full slice of the pizza right n over 360 * pi r^ 2 boom done next we're on to the quadratic formula very important for solving quadratic equations the formula is B plus or- b^ 2 < TK of b^2 - 4 a c all over to a and this is used when you get a quadratic in standard form so for example when it's in ax^2 + bx+ C you plug this in and it's going to give you the value of the X intercepts moreover if this ever equals zero you know that when you use the quadratic formula you're going to get the actual Solutions the values of X next we're on to SOA TOA this is very important for trigonometry okay this is the most important acronym and it stands for S is equal to opposite over hypotenuse and notice that s of a so it depends which angle we're talking about but s of a is equal to the opposite leg divided by the hypotenuse cosine is equal to adjacent over hypotenuse notice the color coding and last but not least tangent is equal to opposite over adjacent and a very important thing to remember is when you have similar triangles meaning the triangles have the same angles and proportionate side lengths sign of corresponding angles in the different similar triangles are always going to be equal next we're on to probab ability which is the number of favorable outcomes over the total number of outcomes so we can quickly say right if you're playing a game and there's 10 different outcomes and there's three of those outcomes that caus you to win your odds of winning are three out of 10 that's basically the gist of probability next we're on to the circle equation very very important they love to ask questions about circles and the circle equation on the coordinate plane so check this out it's x - h^2 + y - k^2 = r^ 2 where if we look at this Circle H comma K is the center of the circle and R is the length of the radius so if we look at it an example here right x - 2 2 + y + 5^ 2 = 36 the center is going to be pos2 5 again we're taking the Opposites of those values and the radius is the square root of 36 which is six next we're on to exponential growth okay so this is our general formula y = a which is our initial value * 1 plus orus r to the the T power again there's our initial value what we're starting with r is your growth rate and T is your time okay and it's plus or minus because this can represent both growth and Decay so when we talking about exponential growth here's an example we have initial value of 200 we're growing at 133% per year and we want to know what is it what is it going to grow to in 13 in three years so we plug and chug but again percentages we enter in decimal format so it's 0.13 that's how you plug and chug now if we have exponential decay it's very similar right we have an initial value now we have a Decay rate it's decreasing at 9% per year and our time is two years here right so instead of adding that 9% we're simply subtracting it AKA what's going to be inside at the end of this is going to be 0.91 that's how you do it done next we're going to talk about the formula to find the vertex of a parabola when we are in standard form again there standard form yal ax2 + BX + C now how do we get the vertex it's simple x = b/ 2 a now this gives you the x value of the vertex to find the corresponding yv value of the vertex you take that value and just plug it back in to the equation number 14 is vertex form of a parabola okay you need to recognize this so this is again standard form down here at the bottom and this is vertex form okay when it's in this form it's really cool because you can tell what the vertex of this Parabola is right and the vertex is going to be H comma K notice it's x minus H so we're taking the opposite of that value right you think it's a negative we flip it and the K we leave alone we don't flip so that's how you do it next we're on to the Pythagorean theorem again that's on the reference sheet but we're going to State it again because it's so important this is A2 + B2 equal c^2 and that's true for any right triangle okay so here's a right triangle We've Got A and B which are the legs and it doesn't matter which is which right B doesn't have to be longer it doesn't have to be shorter it doesn't matter but the key is that c must be that hypotenuse meaning the longest side opposite the 90° angle next round to 16 special right triangles again this is on the reference sheet so you don't have to memorize these but I wanted to show you them because they're very very important and we have two special right triangles a 45 45 90 notice you have two 45 degree angles and a 90° angle and then we have a and by the way for this one the side lengths uh the legs are both the same they're both equal so we call them x and x and then whatever those legs are the hypotenuse is those legs time radical 2 so it's x * < TK of 2 now if you're given the hypoten and you need to find the legs you should of course divide by Rad two we also have a 30 6090 right triangle right where we got the 90° and the 30 and a 60 now in this case the smallest side is X then we go to the uh the leg the bigger leg and that's just the smaller side times radical 3 and then the hypotenuse is double the smaller leg now if you're ever trying to figure out all three sides or an additional unknown side make sure no matter what side you're given to start with you find the the smallest side first so for example if you're given the hypotenuse and you're trying to find the long leg divide by two find the little leg then multiply by red three to get the long leg next we got this nice little formula and you might say distance equals rate time time that's pretty simple but I think it's a very important formula to keep in mind of especially for rate problems okay so distance equals rate times time rate is again is also kind of synonymous with speed and make sure that the units of the rate are the same as time so for example if you've got miles per hour and then they give you time in minutes you need to convert that time to hours next we're going to talk about a relationship between s and cosine that is tested very frequently on the SAT which is that s of an angle equals cosine of the compl of that angle so what do I mean I mean that s of 10 equals cosine of 80 because 10 and 80 add to 90 that's the definition of complimentary angles furthermore s of 20 cosine of 70 s of 30 equal cosine of 60 and so on and so forth right we have all these relationships so just remember that because it's tested a lot next we are going to go over the sum of solutions of a quadratic again this is a Nifty formula because it's tested a lot so if we if they say Hey you know we got this quadratic what are the sum of the solutions so it's not like find a solution or find both Solutions we want the sum of the solutions of course you can find the individual Solutions and add them together but this is a Nifty little shortcut it's simply negative B over a that's how you do it so for example if we have this equation the sum of the solutions would be -7 over pos2 boom done another one that is now being tested more frequently on the digital sat is the product of solutions of a quadratic so similar to the sum of solutions right if we have our standard form here it's going to be C over a now in this example here 2x^2 + 7 x - 5 again C is going to be -5 and a is still two so the product of our solutions would be -5 over 2 now we're talking about the the discriminant again of a quadratic what is the discriminant it's the part of the quadratic formula under the radical and if we have a nice quadratic like this it is B ^2 minus 4 a c and this tells us a lot about this nature of the solutions of a quadratic equation like we see here so for example if we have this as our equation B ^2 - 4 a c would be negative right 9 - 40 31 and when it's negative it means we have no real solutions basically imaginary Solutions and we also have this example now if I take the discriminant here plug it in b^2 - 4 a c it equals zero this means we have one real solution last but not least when it is positive when it is greater than zero we know that we have two real solutions next we have the area of an equilateral triangle okay so here's an equilateral triangle all the sides are equal they all equal s so here is the formula it is s^2 * radical 3 / 4 so for example if all the side lengths are 1 1^ 2 * rad 3 over 4 if we have side lengths of 2 2^ 2ar which is 4 * rad 3 over 4 and last but not least three we'd have 3^ squ which is 9 red 3 over 4 that's how you do it done next we have a series of Pythagorean triples that are great to know and to memorize because it can help you not essential because you can always use Pythagorean theorem to figure out the other side lengths of a triangle but this saves you some time so it's pretty cool most common one is 345 and there's also multiples of 345 that is tested we're going to come back to that we also got 51213 72425 and 8151 17 so these are very commonly tested right triangles on the SAT so the multiples of course would be 6 810 91 1215 etc etc then we got multiples of 5 12 13 just to be mindful of and for the 72425 815 17 I'm not saying to memorize all the multiples I'm just saying keep in mind that if it's a 3 four five you might see it showing up as a 9 12 15 but it's still a 345 next we're going to talk about perpendicular slope okay so if we know that a line has a slope of M right let's call it A over B perpendicular slope is going to be the opposite reciprocal meaning negative B over a and by the way I didn't put it on here but parallel slope means the slopes are equal they're the same okay so if we had a parallel slope to A over B it would just be a over B so here we have an example of 2/3 that is the slope of this line so a perpendicular slope would be -3 another formula to be mindful of is the sum of angles of any polygon and that is equal to nus 2 * 180 where n is the number of sides of any polygon so for example if you take a triangle three sides plug and chug we get 180 right but that's one that one is actually on the reference sheet so you don't need to memorize that then we have a quadrilateral we got four sides 4 - 2 * 180 is 360 and here's one more example for a pentagon 5 - - 2 * 180 is 540° next we have rational functions which are now being tested more so on the digital sat than they were previously and so a rational function is something like this right something over something x - 4 over x + 3 so what we need to know is what we can figure out from this equation right here so I'm going to take two examples and we're going to look at their graphs right so when we look at the graph of the function we'll notice that the x intercept is at 40 whatever is going to zero out the numer Ator and there's an ASM toote vertical ASM toote and whatever zero is out the denominator so in this case it's going to be x = -3 lastly we can also quickly figure out what is the Y intercept by simply plugging in zero for X which gives us -4/3 okay now we'll do the same thing on the right uh to find the x intercept what Zero's out the numerator is going to be -2 where's our vertical ASM toote it's going to be at positive five right x equal positive 5 and last but not least plug in zero for x and we get ne 25ths for our Y intercept next we want to know all of our triangle congruence theorems okay all the way that we can prove triangles are congruent and we got five of them we got side side side show that all the sides are congruent triangles are congruent we got side angle side and very important the angle has to be nested between those two congruent sides right it has to be in that specific order to prove congruency we've also got angle side angle same deal that side has to be nested right in between those two congruent angles angle angle side and last but not least hypotenuse leg that's only for right triangles but if we show that the hypotenuse are equal and one pair of legs are congruent the triangles must be totally congruent and again just in case you don't remember what that means that if triangles are congruent in means every corresponding angle is equal and every corresponding sides are equal we also want to know our triangle similarity theorems and there's three of them okay and proving triangles similar means that they're not congruent but they're proportionate okay it's kind of like a Xerox image blown up all the dimensions are proportionate so we have the most commonly tested one which is angle angle show that any two angles in two triangles are congruent they must be similar they must be proportionate we also have side angle side now it's different from side angle side in congruency because we're saying side angle side I'm not saying that the sides are congruent I'm saying that we've got congruent angles and the sides around those angles are proportionate okay so if there's a 1 to two ratio for AC and XZ that 1:2 ratio has to exist for ab and XY as well last but not least we've got side side side again not all the sides are congruent but they're all proportionate there's a consistent proportion so one way to think about this is like that idea of a scale factor right ABC looks like the smaller one than XYZ here so we could imagine that every side of ABC is half the length of XYZ and that would show that they are indeed proportionate and similar by side side side next we have the unit circle so the unit circle is not essential but it's recommended and it can save some time if you if you do have a decent understanding of it and how it works so for example here's here's the unit circle and if we are looking at what the sign of an angle represent in the unit circle it represents the Y value okay of these coordinates I'm going to show you what I mean as we go through an example cosine is the x value and then we have tangent of theta is equal to y overx so what am I talking about here so we're going to say that for example s of 45 is equal to rad 2 over 2 why where do where does that come from because if we look on the unit circle that is the yvalue of that coordinate at 45° what about cosine of 150 that's negative red 3 over2 why because again look at that coordinate at 150° the x value of that coordinate is negative red 3 over2 last but not least we got tangent of 240 that's rad three why is that because again if we look at tan uh 240 the ratio or you know rad 3 over2 / - 1/2 that simplifies to rad 3 that's how you do it now this one is more of an algorithm really than a formula but it is all about foiling okay and when I'm talking about foiling I'm talking about foiling to binomials so this stands for first outer inner last okay and you can see everything popping up on the right hand side as I'm going through those so let's carry this out right so when first means we're going to multiply those first two terms x * X and that is going to give us x^2 next we're going to multiply the outer terms in red that's going to give us -6 * X -6x then we're going to multiply the inner terms 3 * X which gives us 3x and then the last terms 3 * -6 -8 we're going to combine those two terms in the middle because they're like terms -6x + 3x and we're left with x^2 - 3x -8 squaring a binomial this is just like foiling and and reason why I want to point this out is because it's a place where people can make a lot of silly mistakes so for example this is a binomial that's being squared so you might look at it and be like oh cool it's just x^2 + 9 right you're squaring each of them but we don't do that we don't distribute the exponent like that this is incorrect when we're squaring a binomial we're literally multiplying it against itself so now we're back to that whole idea of foiling right we're doing the first two terms then which is x * X which is x^2 the outer terms right which is 3 * X the inner terms which is again 3x and the last terms which are which is 3 * 3 which is 9 again combining like terms 3x + 3x gives us 6X in the middle there's your final answer done next we are on to exponent rules these are super super important so we're going to go through all the critical exponent rules that you have to know on the SAT all right first of all when we multiply the same bases we add the exponents okay and when we have this kind of floating exponent uh outside of parenthesis we multiply it in right so that becomes x to the m * n over here we have division that is going to with the same basis again we're going to subtract the top minus the bottom when we have something raised to a negative exponent it's really going to be 1 /x to that M Power anything raised to the zeroth power is going to be one and anything raised to the first power is just itself and last but not least this is a rational exponent turning into its radical form the Top Value is the power so it's x to the m the bottom value the denominator is the index and by the way you can take the index first and then raise it to the exponent you can go either way now we're going to go through numerical examples of each so for example 2^ 2 * 2 the 3r we add the exponents and we get 2 to the 5th which is 32 here's another example same numbers but now we're going to multiply those exponents and we get 2 to the 6th which is 64 uh now we have an example again same basis we're going to subtract those exponents 5 - 3 is 4 squared which is of course 16 now we have 5 to the -3rd again that just simply means we're dropping it to the denominator putting a one on the top 5 to the 3 power is 125 so it's 1 over 125 here we got anything to the zeroth power here's 7 to the zero is going to be one anything to the first power is just itself AKA 9 and last but not least we got 8 to the 2/3 so let's let's check this out let's break it down it's going to be the cube root of 8 SAR what's 8 squared it's 64 what is the cube root of 64 4 because 4 * 4 * 4 is 64 now we could have also done it the other way the cube root of 8 is 2 2^2 is four so in either case we're getting four next I want to talk about the basic protocol for solving an absolute value equation okay so let's say we have x - 4 is equal to 6 we should get two solutions not one and the way we do it is we split it into two equations the first equation is exactly the same we just remove the absolute value bars the second equation is almost exactly the same but we put a negative on that constant value outside of the absolute value so we flip it now we solve these equations as is we're going to add four to both sides and x equals 10 on this one same thing add four to both sides and we get xal -2 those are our two solutions done last but not least we have a little thing that I want show you on inequalities that I want to make sure you don't forget okay let's say we have this inequality and you need to solve it all right we're going to isolate X so we're going to add four to both sides no problem then we got -3x is less than 9 divide both sides by -3 okay standard protocol and we've got X is greater than -3 what happened to the inequality it flipped it was less than now it's greater than why do we do this this is the rule to remember don't forget to flip the inequality symbol when dividing or multiplying both sides by a negative so just bear that in mind very important if you do forget it you will make a mistake on this test all right guys that is it for the digital sat math formula Bible I hope you enjoyed this video I hope you crush this test when you take it and I wish you all the best of luck on your SAT Math Journey if you like this video make sure to click that like button if you want to see more from the scaler Learning Channel make sure to click subscribe thank you guys so much for joining and I'll see you in the next video take it easy