now in this lesson we're going to solve quadratic equations by completing the square so let's start with this one x squared plus 4x is equal to 5. so to complete the square take half of the number in front of x which is 2 half of 4 is 2 and square it so we're going to add 2 squared to both sides whatever you do to the left side you must also do to the right side now to factor this expression x squared plus 4x plus 4 it's going to be x and then whatever this sign is plus this number before you square it x plus two squared if you factor x squared plus four x plus four it's x plus two times x plus two two numbers that multiply to four and add to four is two and two on the right side we have five plus two squared which is four and five plus four is nine so now let's solve for the variable x let's take the square root of both sides the square root of x plus two squared is x plus 2 and the square root of 9 is plus or minus 3. so now we have two equations x plus 2 is equal to 3 and x plus 2 is equal to negative 3. so if we subtract both sides by 2 we can see that x is equal to one that's one of the answers and for the other one if we subtract both sides by two negative three minus two is negative five so x is equal to negative five now let's work on another example try this one 2x squared minus 12x minus 7 equals zero solve by completing the square so first i'm going to move the negative seven to the right side so two x squared minus twelve x is equal to positive seven now notice that there's a 2 in front of x squared what we need to do is factor out the 2. if we take out 2 it's going to be x squared minus 6x and then i'm going to leave a space now let's complete the square half of six is three and we need to square it so i've added not just nine to the left side but nine times two which is eighteen now whatever you do to the left side you must also do to the right side so i have to add positive 18 to the right side now to factor x squared minus 6x plus 3 squared it's going to be x whatever this symbol is minus this number squared x squared minus six x plus nine is x minus three times x minus three and then seven plus eighteen that's 25 now let's divide by two so x minus three squared is equal to 25 divided by two now we can take the square root of both sides so on the left it's just going to be x minus 3. on the right plus or minus the square root of 25 is 5. the square root of 2 we'll just leave it as root 2. now let's rationalize this expression let's multiply the top and the bottom by root two so x minus three is equal to plus or minus five root two over two now all we could do now is just add three to both sides so x is equal to three plus or minus five root two over two now we can't add three and five root two over 2. we can't add them together so we're just going to leave it like this this is the answer now keep in mind there's two answers the first answer is 3 plus 5 root 2 over 2 and the second answer is 3 minus 5 root 2 over 2. now let's try another example three x squared minus five x is equal to ten solve by completing the square now the first thing we're going to do is take out a 3 from the first two terms 3x squared divided by 3 is x squared negative 5x divided by 3 well it's just going to be negative 5 over 3x now we need to take half of this number half of 5 over 3 is basically 5 over 6. multiply the denominator by 2. that's how you can find a half of a fraction and then once we take half of it we need to square it so it's going to be 5 over 6 squared now on the left side we've added 5 over 6 squared but time stream so we got to add that to the right side as well now how can we factor this expression everything you need you can see it it's going to be x and then whatever this sign is minus this number before you square it 5 over 6 squared it's always going to work out that way on the right we have a 10 plus 3 5 squared is 25 6 squared is 36. now let's combine this first 3 over 36. if we divide 36 by 3 we're going to get 12. so this is the same as 25 over 12 plus 10 over 1. now let's combine 10 of 1 and 25 over 12. to do that we need to get common denominators let's multiply this by 12 over twelve so that becomes one twenty over twelve one twenty plus twenty five is 145 over 12. now what we need to do is get rid of this three so we're going to multiply both sides by one third on the left side three and one-third will cancel on the right side we might be able to reduce a few things well it turns out that 145 is not divisible by 3. so we can't reduce that so let's multiply 12 times 3 which is 36 now let's take the square root of both sides we can't really simplify the square root of 145 so we'll just have to leave it the way it is but the square root of 36 is six and this is going to be plus and minus all we can do is now add this fraction to this side so the final answer x is equal to 5 over 6 plus or minus root 145 that's a terrible five divided by six so this is the answer if you want you can write it like this too as a single fraction since we have the same denominator 5 plus or minus root 145 over 6. you can leave it like that if you want to