here are the top 10 most important things you must know about quadratics must know number one what is a quadratic relationship let me show you how you can recognize a quadratic relationship from the three ways any relationship can be represented from an equation from a table and from a graph an example of an equation that involves a quadratic relationship between an X and A Y variable is y = x^2 - 6X + 8 this is a quadratic relationship because the highest exponent on the variable is two quadratic relationships are also known as degree2 polinomial equations now let me show you a table of values for this quadratic relationship and we'll analyze the properties of that table I'll choose some X values and now I'll calculate y by taking each of these X values summing them in for X into the equation and getting the value for y from this table you should be able to tell that the relationship between X and Y is quadratic and how you could tell that is by analyzing the finite differences of the Y values if I look at the difference in the first pair of Y values always do the bottom one minus the top one so 0 - 3 is -3 and then do that for each pair of Y values notice they're not constant so it's not a linear relationship but if I were to find the difference in each of these pairs notice that the second column of finite differences is constant and that's going to be true for any quadratic relationship and now I'll plot each of these points from this table and we'll see what the graph of a quadratic looks like if I connect these points I get this u-shaped graph and that u-shape we call it a parabola all quadratic relationships form that Parabola shape and these Parabola shapes have a couple important characteristics all parabas have a Vertex which is its Max or m point so I'll label it in green here here's the vertex of this Parabola and another important feature is all parabas are symmetrical about a vertical line that goes through the vertex and we call that the axis of symmetry and now you know how you can recognize a quadratic relationship from an equation table and graph the second thing you must know about quadratics is the standard form equation of a quadratic the standard form equation of any quadratic relationship takes the format y equals ax^2 plus BX plus C where a b and c could be any real number from a standard form equation we can tell a couple properties about a quadratic the a value will tell us the direction of opening if a is greater than zero I know the parabola opens up and if a is less than zero the parabola opens down another important feature of the standard form equation is the C value the C value reveals the Y intercept of the quadratic the Y intercept will be at0 comma C and now let me give you an example of this let's say we have the quadratic y = 2x^2 + 4x - 5 in this quadratic the 2 is the a value 4 is the B value and C the constant is5 what do these tell me about the quadratic well since the a value is two that's bigger than zero so this quadratic opens up and the C value is5 so I know the Y intercept is at 05 if I wanted to see what the graph of this relationship looked like the best way to graph it when you're given the standard form equation is just to make it table of values you can choose your X values and then calculate your y values and you'll get good as you learn more about quadratics at picking good X values which will give you the good Parabola shape of the quadratic calculate the Y's just by subbing all of your X's into this equation and then plot them to see the graph now you know the standard form Basics must know number three vertex form Basics there's another format that the equation of a quadratic relationship can be written in it's called vertex form and it looks like this y = a * x - h^ 2 + K the important thing to know about this format of a quadratic equation is that the H and K values tell you the coordinates of the vertex so off to the side here I'm just going to sketch a random Parabola and its vertex would be at h k the axis of symmetry is always the vertical line that goes through the vertex so the equation of the axis of symmetry is going to be x equals the x coordinate of the vertex which is H so the equation of the axis of symmetry is x equal H and lastly just like the standard form equation the a value tells us the direction of opening if a is greater than zero the problema opens up and if a is less than zero the par opens down now let's say we had a quadratic that was in standard form and you were interested in the vertex of that Parabola what you could do is you can convert it to vertex form through a process called completing the square and let me give you an example of that let's say we had the quadratic y equal -2X 2 + 20x - 11 in order to get that into vertex form you put the first two terms into a group you common factor out the coefficient of the X squ from the first two terms only and then I left spaces here for the number that we have to add and subtract that quadratic in the brackets to turn this into a perfect square trinomial it'll be a perfect square trinomial if the constant at the end is half of the B value squared so half of 10 is 5 and 5^ squ is 25 so I'll add and subtract 25 to keep it equivalent get the -25 out of the brackets by multiplying it by the -2 in front and then factoring the quadratic that's in the brackets should be easy because it's a perfect square trinomial the numbers that multiply to the C value and add to the B value will be the same number twice it' be5 and5 so it would factor to x - 5^ 2ar and now that it's in vertex form the H and K values tell you the vertex notice from the vertex form equation H is what we're subtracting from X so in this equation we're subtracting five so the H value is five so the vert X is at 539 and the axis of symmetry would be at x = 5 and because the a value is negative this probab is going to open down which means this vertex is at the top of the quadratic making it a maximum point and I can make a rough sketch of that quadratic for you the vertex is at 539 from the standard form equation I can tell the Y intercept is at ne1 and I know this Parabola opens down now I know the basics of the vertex form of a quadratic must know number four factored form Basics the last format of a quadratic equation you need to know is the factored form and it looks like y equals a * x - M * x - n let me make a rough sketch of a random Parabola and we'll talk about what the factored form tells you about that Parabola since the factored form is a a product of factors if we want to know the X intercepts of a quadratic well we know what the X intercepts Y is zero so we would just figure out what makes any of these factors be zero x - M would be Z if M was Z and x - n would be zero if n was zero so if x was either M or n this whole product would be zero meaning it's an X intercept so the X intercepts are at M and N and then because parabas are symmetrical about the axis of symmetry I know that if I found the average of M and N that would give me the axis of symmetry so the equation of the axis of symmetry would be x = m + n / 2 and lastly of course the a value tells us the direction of opening of the parabola let me give you an example of a quadratic that's in factored form the most useful part of factored form is it's easy to find the X intercepts I know at any x intercept the Y value is zero so I could set y to zero and then using the zero product rule I know the product of these three factors is zero if any of them are zero x + 2 could be Z if x was -2 and x - 6 could be zero if x was 6 so the X intercepts are at -2 and 6 let me make a rough sketch of what we have so far now if I was interested in the axis of symmetry I know it's going to be right in the middle of these X intercepts I can find the middle of -2 and 6 by adding them and dividing by two so the equation of the axis of symmetry would be x = -2 + 6 / 2 which is 2 now the vertex of this paraba is going to fall somewhere on this axis of symmetry so I know it's x coordinate is going to be two and to calculate the y-coordinate of the vertex we could just take this value of two and sub it in for both of the x's in the factored form equation and if you do that you would get a y-coordinate of8 so the graph of this quadratic looks something like this there you go now you know the factored form basics of quadratics must know number five you have to know how to factor quadratics and we Factor quadratics as a means of getting a standard form quadratic converted into factored form and let me give you three examples of typical quadratics that you may need to know how to factor let's look at this first quadratic when the quadratic in standard form has a leading coefficient of one so an a value of one we can do a product and sum method of factoring this quadratic where we find numbers that have a product of the C value and a sum of the B value the pair of numbers that multiply to 24 and add to 11 are 8 and 3 so what we do with those numbers to convert the standard form quadratic into factored form we make two factors the first term of each of those factors is X and then each of those X's I add the pair of numbers that satisfy the product and sum so I take the eight put it here and I take the three and put it here and there we have it there's the equivalent factored form of this standard form quadratic the next type of quadratic you have to know how to factor is one where the a value is not one when that happens we're still going to find a product and sum but this time we're going to find numbers who have a product of a * c 2 * -4 is8 and a sum of the B value 7 the numbers that satisfy that product and sum are 8 And1 now what we do with those numbers is different than in our first example we actually split the middle term into a sum of 8X and -1x so I'll rewrite the original quadratic equation as 2x^2 + 8 x - 1X - 4 notice all that happened was we rewrote that positive 7x as 8x - 1X it's equivalent but this will allow us to now Factor this by grouping if I look at the first two terms if I take go to common factor I would take go to 2X and the last two terms I can take out a common factor of1 and now both of these terms have a common binomial of x + 4 so that can be factored out and once it's factored out we're left with 2x - 1 as the second factor and the last thing I want to show you is a common trick that will show up and it's if we have a difference of two perfect square values if we have that there's a rule it's called the difference of squares rule if you have an A S minus a b squ squ it factors to a minus B * A + B to help you see that this is a difference of squares I'm going to rewrite the 9 as a 3^ squared So based on the difference of squares rule I know this would factor to x - 3 * x + 3 there you go now you know how to factor quadratics must know number six you need to know how to solve quadratic equations using the method of factoring so I'm going to give you two equations that we can solve by factoring let's look at the first equation I want to figure out what value of x could I sub into that equation to make the left side equal to the right side since I want to know when this is equal to zero I know if it was in factored form it would be easy to find out what would make that product be zero so I can get it into factored form first by removing a common factor of two from all the terms and now looking at the quadratic and brackets I could Factor it by finding numbers that multiply to C and add to B the numbers that multiply -12 and add to -1 are -4 and 3 so I could factor that quadratic that's in Brackets into x - 4 * x + 3 and now using the zero product rule I know that the product of those three factors is zero if any of the three factors are zero well x - 4 would be 0 if x was 4 and x + 3 would be 0 if x was -3 so there are two answers to that equation 4 and-3 the second equation notice that we're trying to figure out how can the left side equal 10 that's a little more difficult so the first thing we'll do is take that 10 and move it over you should always set your quadratic equal to zero before solving and now I'll solve this by factoring because the a value is not one I'll need to find numbers that have a product of a * C so a product of -30 and a sum of the B value 13 the numbers that work are 15 and -2 so I have to split this middle term into 15x - 2x and now I factor by grp grouping I'll remove a common factor from the first two terms I'll take out a 3X and I'll take out a common factor from the last two terms I'll take out a -2 I have a common binomial of x + 5 I'll factor that out and when I factor that out from both terms I'm left with 3x - 2 as my second Factor now that it's in factored form I know that product would be zero if either the x + 5 was zero or if the 3x - 2 was Zer x + 5 would be Z if x was5 and 3x - 2 would be 0 if x was was 2 over 3 so once again two answers to that quadratic now you know the basics of solving by factoring must know number seven solving by completing the square let me give you another equation to solve now not all quadratics can be solved by factoring for example there are no numbers that multiply to8 and add to -6 so for that reason we need a different method for solving this quadratic if we convert this to vertex form by completing the square we'll be able to easily rearrange to isolate X so remember to complete the square we put the first two terms in Brackets and then we want to create a perfect square trinomial by inside the brackets adding and subtracting half of the middle term squared half of six is three and when we Square it we get N9 I'll get the9 out of the brackets by multiplying it by the one that's in front and then this quadratic is a perfect square trinomial so it factors easily the numbers that multiply to 9 and add to -6 are -3 and -3 so it factors x - 3^ 2 and now we just rearrange this to isolate X I'll start by moving the -7 to the other side and then I'll move the square to The Other Side by doing the inverse of squaring which is plus or minus Square rooting so x - 3 equal plus or minus the square otk of 17 and then lastly move this three to the other side so I have x = 3 plus or minus < tk7 there are two answers there 3 +un 17 and 3us < tk7 both irrational answers so I'll leave it in exact form just like this you could use a calculator to get an approximate value for each of those answers if you wanted to there you go now you know completing the square is not only a method for figuring out the vertex of a quadratic but it can also find you the solutions to a quadratic equation must know number eight solving using the quadratic formula now if we had just a general standard form quadratic and if you apply app the method of solving by completing the square to that General quadratic it would end up rearranging to be xal B plus or minus theare < TK of b ^ 2 - 4 * a * C all over 2 * a that's the quadratic formula and it can find you the solutions to any quadratic equation let me show you an example let's actually use the quadratic equation that we had in the last example we know the answers to this should be 3 plus orus < 17 but we can avoid the whole process of solving by completing the square by just subbing into this quadratic formula the a b and c values of this quadratic that has been set equal to zero subbing into quadratic formula it tells me to do the negative of the B value so -6 is just 6 plus or minus the Square t of the B value squared so - 6^ 2 - 4 * the a value time the C value that's * 1 * 8 and that all needs to be divided by 2 * the a value of 1 if I simplify underneath the square root I have 36 minus -32 that's 68 this radical can actually be simplified to the < TK of 4 * the < TK of 17 and I know theun of 4 is just two so I'll rewrite it as just 2un 17 and and now in this fraction the 6 is being divided by two and the 2un 17 is being divided by 2 that means I could simplify this 6id 2 is 3 and 2un 17 / 2 is just < tk7 notice those are the same two answers we got by solving by completing the square this just avoids us having to do that whole process those are the exact answers you could get the approximates just by typing on your calculator 3 +un 17 and 3-7 now you know how quadratic formula Works must know number n you have to know about the discriminant of a quadratic the discriminant if I show you quadratic formula is the part underneath the square root the B ^2 minus 4 a c whatever you get underneath the square root in quadratic formula actually tells you a lot about the number of solutions or the type of solutions you're going to have to the quadratic if b ^ 2us 4 AC is bigger than zero you will get two real solutions to the quadratic those two real solutions could be rational or irrational numbers they will be rational if the B ^2 minus 4 a c value is a perfect square number but if b^2 minus 4 a is not a perfect square value you'll get irrational Solutions another useful tip is that if b^ 2us 4 a is a perfect square that also indicates that the Quadra is factorable but if b^2 - 4 a c is not a perfect square then you know the quadratic is not factorable if B S - 4 a is equal to zero you will only get one real solution to the quadratic equation and that's because if underneath the square root is 0 the square root of 0 is 0 so you'll be adding and subtracting 0 from B adding and subtracting Z gives you the same value therefore only one solution if B ^2 - 4 a c is NE so if it's less than zero there are no real solutions to the quadratic equation there are complex Solutions but no real solutions so what could all of these scenarios look like graphically for the quadratic if a quadratic has two Zer you would know that it intersects the x axis twice if it has one zero you know that its vertex must be right on the xais and if it has no zeros that means means the vertex is above the xaxis and the problem opens up or it's below the x-axis and the quadratic opens down and there you go now you know the usefulness of the discriminant of a quadratic must know number 10 you need to know a few different ways of finding the vertex of a parabola let's say we had the standard form quadratic y = 2x^2 - 12x + 10 if I wanted to find the vertex of that Parabola there's a few different ways we could do that we could do completing the square we could average the X intercepts or a shortcut to that we could do B over 2 a let me show you all three of those so remember completing the square is a method of getting that into vertex form first two terms in Brackets common factor out the two that's in front of the x s then add and subtract half of 6 which is 3 which is 9 so add and subtract 9 get the9 out of the brackets by multiplying it by the two in front and then lastly factoring the quadratic in the brackets it factors to x - 3^ 2 and outside the brackets I have A8 so the vertex is 38 let's see another way we could find that by first finding the X intercepts of the quadratic so I'll set y to zero and let me common factor to two while I'm at it and now I'll try and Factor this quadratic by finding numbers that multiply to 5 and add to -6 the numbers that satisfy that product and sum are -5 And1 I know this product would be zero if any of the factors are zero x - 5 could be zero if x is 5 and x - one could be zero if x is 1 now I know the vertex is going to be right in the middle of the X intercepts so the x coordinate of the vertex I just have to average five and one add them and divide by two and I figure out the x coordinate of the vertex is three and then to find the y-coordinate I can just use the standard form equation of the parabola just replace each of the X's with three and you'll get a y-coordinate of8 or a shortcut to all of this you can just use the A and the B value of the standard form quadratic to find the x coordinate of the vertex all you have to do is this formula here b/ 2 a so -12 which is 12 over 2 * the a value so over 2 * 2 so I have 12 over 4 which is 3 and then once you have the x coordinate you can just calculate the Y by subbing in the X into your equation and you would get ne8 so there you go three different ways of finding the same vertex of that Parabola hopefully these 10 must knows made you feel more confident with quadratic relation ship let me know what you want a top 10 of next [Music]