[Music] welcome to solve equations so this video all we're going to do is solve equations now they're going to be some higher order equations some linear equations some quadratics just different kinds so hang on here we go an equation in X is a statement that two algebraic expressions are equal so in other words we're going to have one expression on one side of an equal sign and one expression on the other to solve means that we're going to get all the x's on one side and all the numbers on the other and the solution will be x equals to a number it is true if we have that statement and that means there is one solution an identity is if we cannot if we get 1 equals one and that's when we check a solution an equation that is true for just one solution in real numbers or for a set of solutions sorry that is true for a set of real numbers is called a conditional equation we're not going to deal with any of those in this section but we'll come up with those later so how would we approach solving negative three parentheses 4 T minus five close parentheses equals 5 parentheses 6 minus two T close parentheses so to solve this the first thing we're going to do is we're going to distribute the negative 3 on the left and the 5 on the right which gives us minus 12 t plus 15 equals Thirty minus 10 t now we want to subtract 30 from the right and add 12 on the left that will move all the t's to the right and all the numbers to the left now usually when you see this you see people write it vertically I've written it horizontally just to save a little bit of space when we do all the out do the simplification what we end up with is minus 15 equals 2T the minus 12 T and the plus 12T cancel we get 15 minus 30 that gives us minus 15 that's the left side on the right side we have 30 minus 30 minus 10t plus 12T gives us a positive 2T now we're going to divide by 2 on both sides which gives us the final solution of negative 15 over 2 equals to T next how would we solve 2x plus 2 divided by x minus 2 equal to 1. the first thing we want to do is multiply both sides by the LCD that is the least common denominator which is x minus 2. because 1 does not have a denominator it would be actually 1 over 1. so when we do this we have x minus 2 times parentheses 2x plus 2 over x minus 2 close parentheses equals 1 times x minus 2. on the left side those x minus twos cancel out and leave us with 2x plus 2 on the left equal to x minus 2 on the right now what we're going to do is we're going to subtract X from the right and subtract 2 from the left so that moves all the numbers to the right side and all the letters to the left side when we do this we get 2x minus X plus 2 minus 2 equal to x minus X minus 2 minus 2. which simplifies down to x equals negative 4 which is our final solution because this equation is a fraction we need to check our answer and make sure that x minus 4 does not produce zero in the denominators to check the solution we're going to have to put X equal to 4 everywhere there's an X in and then make sure that the left side equals to 1 on the right side so we're going to have 2 parentheses negative 4 plus 2 over negative 4 minus 2 and we want that whole quantity to become 1. we're not going to do anything with the right side but we're going to simplify down the left side and make sure we get 1. using our order of operations we see that the numerator that 2 times negative 4 becomes negative eight and that the denominator we get minus 4 minus 2 is minus 6. now we can work those two as separate Expressions because ones in the numerator and one's in the denominator now for the next step we're only going to work with the numerator because the denominator is already simplified so we have minus eight plus two that simplifies to negative six the denominator is already negative six again we're leaving the right side alone we have negative 6 over negative 6 which is one so we have one o equals to 1 which tells us that x equals negative 4 is the correct solution to our original equation now we want to solve x cubed minus 4X equals to 0. how are we going to go about this well guess what we get to do my favorite thing which is factoring so the first thing we're going to do is we're going to pull out the greatest common factor which is X so when we do that we're left with x times x squared minus 4 equal to 0. now how do you factor X squared minus 4 remember 4 is a perfect square so now we're going to use that difference of squares formula which is a squared minus B squared equals a minus B times a plus b this gives us x times x minus 2 times X plus 2 equal to 0. now what we're going to do is we're going to set each of those factors equal to 0 and solve so we have x equals to 0. there's one solution x minus 2 equals to 0 which tells us adding 2 to both sides that we get x equals to 2 X plus 2 equals to 0 subtracting 2 from both sides we find that x equals negative two so our three solutions are x equals zero x equals two and x equals negative two now our La our next equation is 1 minus nine over x squared equals zero so what are we going to do first if you have a fraction then what you want to do is multiply every term by your least common denominator so your least common denominator is x squared the denominator of one is one the denominator of zero is one that makes our else our g c f our greatest or least common denominator that should be LCD not GCF so let's just Mark through that real quick that should be l c d is x squared so then we're going to multiply x squared times 1 minus x squared times nine over x squared equals 0 times x squared now doing this what we notice is that these X squares cancel out so what that gives us is the equation x squared minus 9 equals 0. now remember 9 is a perfect square so we get to use that difference of squares formula again so we have a squared minus B squared is a minus B times a plus b so we have x minus 3 times X plus 3 equals 0. next step set each of those factors equal to zero and solve so we get x minus 3 equals 0 add 3 to both sides so we get x equals 3. set X plus 3 equal to zero so we get X then subtract 3 from both sides and we get x equals negative 3. so our Solutions are x minus 3 and X Plus 3. okay what happens if we want to solve x plus eight equals to 1 over x minus 4. again we're going to multiply both sides by our LCD which is x minus 4. X only has a denominator of 1 and 8 has a denominator of one but this time I'm going to treat that X Plus 8 as one term not as two so now my x minus 4S on the right are going to cancel so now what I need to do is expand the left side so this should be expand the left side to solve the quadratic I have to expand that left side because it's equal to 1 but in order to solve a quadratic I have to have it set equal to 0. so that means I have to multiply that quadratic out all right so let's expand the left side to solve the quadratic I have to set it equal to zero so that means that that left side has to be multiplied out and then I have to pull that 1 over to the other side so when we multiply that quadratic out the two factors we get x squared plus 4X minus 32 using foil first outer inner last equals to 1. now we pull that 1 over to the other side so we have x squared plus 4X minus 33 equals 0. now how do we solve that we know it's not factorable because we just factored we just went from a factored form there are no two factors of 33 that will subtract to give me four because the only factors of 33 are 1 and 33 3 and 11. that's it so I have to use the quadratic formula that tells me it's negative B plus or minus the square root of B squared minus 4AC all over 2A equals X plugging my values in b equals negative 4. b equals 4 so negative B is negative 4 plus or minus the square root of B squared which is 4 squared minus the 4 in the formula times a is one and C is negative 33 all over 2 times 1. now plugging that into my calculator and simplifying things down I get negative 4 plus or minus 2 square roots of 37 over 2. going further I can simply break that into two fractions then I get negative 2 plus or minus the square root of 37 are my two roots or my two answers I know there's going to be two answers because the highest power of my polynomial is 2. let me know if you have any questions thank you foreign [Music]