questions one solve the following quadratic equations by using completing the square method give your answer in three decimal places before you start make sure the quantity equation is written in general form which is x squared plus b x plus c equal to zero and a is equal to positive one so look at questions a x squared plus four x minus nine equal to zero so this equation is written in general form and a is equal to 1. therefore we can start solving these questions so first we're going to move this c to the other side become a positive 9. next we are going to add in a number here so that the equations on the left here representing the area of a square here you have a b of a positive four so you're going to divide the b with two so when number four divide by two you're going to have a number 2 then square this number then this part will represent the area of a square on your right hand side you need to plus 2 squared as well so that these equations will be kept balanced next i can factorize this part as this is the area of a square now noted here this is a plus two just copy this part down into the bracket plus two and put a square here therefore you already factorize it this one is nine plus 2 squared which is 13. next you want to move this square to the other side it will become plus minus square root of 13. so x is equal to negative 2 plus minus square root of 13. so we have two value of x one of it is negative two plus square root of thirteen or the other one is negative two minus square root of 13. solve this using your calculator it will give you the answer of 1.606 or negative 5.606 questions b x squared minus three x minus five equal to zero so this equation is written in general form and a is equal to one therefore we can start solving this by moving the c to the other side x squared minus three x equal to five next we're going to add in a number here so for this part this this will equal to the area of a square so here your value of b is a negative 3. divide this with two therefore you're going to have negative 3 over 2 and put a square for this number so this is the area of a square so you need to add negative 3 over 2 squared on your right hand side to keep these equations balanced next you can start factorize for this part on the left hand side so it will be x now here you have negative 3 over 2 just copy this down and put a square therefore you already factorize it so for this part if you solve it you have 29 over 4 next move this square to the other side it will be plus minus square root of 29 over 4 so a x is equal to 3 over 2 plus minus 29 over 4. so we have two value of x one of it is three over two plus square root 29 over 4 or the other one is 3 over 2 minus square root 29 over 4. so solve these two using the calculator it will give you the answer of 4 point or negative 1.193 question c negative x squared minus 6x plus 9 equal to 0. so this equation is written in general form but the a is a negative 1. therefore we are going to divide each of the terms here with negative 1 so that the a of the x squared will become a positive 1. so here you're going to have x square plus 6 x minus 9 equal to 0 then we can start solving this using completing the square method so first move this c to the other side next we're going to find a number to add in here so that this part will represent the area of a square so in this case your b is a plus six divide it by two it will give you the answer of three then square it so on your right hand side you need to add in 3 squared as well so that this equation will be kept balanced next factorize this part noted here this is a plus 3. so just copy this part down here and square so this is the answer this will give you the answer of 18. next move the square to the other side it will be plus minus square root of 18. so x is equal to negative 3 plus minus square root of 18. so you have 2 answer of x one is negative three plus square root of eighteen or the other one is negative three minus square root of eighteen solve this using your calculator the answer is one point two four three or the other one is negative seven point two four three questions d two x square minus six x plus three equal to 0. so this equation has been written in general form but the a is not equal to 1. therefore we are going to divide all of the terms here with number 2. therefore for the d the equations later on will be x squared minus three x plus three over two equal to zero next we can move the c here to the other side then next we're going to add in a number here so that the left hand side the equations will be equal to the area of a square so the b is a negative 3 so divide this negative 3 with 2 then you're going to get negative 3 over 2 and put a square here equal to negative 3 over 2 and then you also need to plus negative 3 over 2 squared on the right hand side to keep these equations balanced next for this part this is the area of the square then we can factorize it noted here this one is a negative 3 over 2 so just copy this down and put the square here and for this part you can just solve it using your calculator it will give you the answer of 3 over 4. next move the square to the other side it will be plus minus square root of 3 over 4 so x equal to so move this to the other side three over two plus minus square root three over four so we have two value of x one is three over two plus square root three over four or the other one is 3 over 2 minus square root 3 over 4. solve this using a calculator you get answer of 2.366 or 0.634 questions d questions e so 4 x square minus 8 x plus 1 equal to 0 so this equation has been written in general form but the a is not equal to one therefore we're going to divide this of the term with number four so for the e next we're going to have x squared minus 2x plus 1 over 4 equal to 0. next move the c to the other side then we're going to find a number here so that this part will equal to the area of a square so the b here is a negative 2 divide negative 2 by 2 then you are going to get negative 1 then square it so on your right hand side you need to plus negative 1 squared as well to keep its balance next factorize this so here you have a negative one so just copy down the negative one here put the square this part solve it using your calculator it will give you the answer of 3 over 4 next move the square to the other side it will be plus minus square root of 3 over 4. so x is equal to move this negative 1 to the other side 1 plus minus square root 3 over 4. so the value of x one of it is 1 plus square root 3 over 4 or the other one 1 minus square root 3 over 4. solve this using your calculator you're going to get 1.866 or 0.134 questions f so negative 2 x squared plus 7 x plus 6 equal to 0. so this has been written in general form but the a is not a one this is a negative two therefore we're going to divide all of the terms here with negative two so for the f later on you're going to have x squared minus 7 over 2 x minus 3 equal to 0. next we're going to move the c to the other side so x squared minus seven over two x equal to three so next we're going to find a number uh to put in here so that this will be equal to the area of a square so your b now is negative seven over two so when you divide it by two then you're going to get negative seven over 4 then square it so on your right hand side you also need to plus negative 7 over 4 squared next factorize this part so here you have a negative seven over four so just copy this down so x minus seven over four and put a square here so for this part solve it using your calculator you will get the answer of 97 over 16. next move the square to the other side so it will be plus minus square root 97 over 16. so x is equal to later on you're going to move this negative 7 over 4 to the other side so positive 7 over 4 plus minus square root 97 over 16. so the value of x is 7 over 4 plus 97 over 16 or 7 over 4 minus 97 over 16. solve this using your calculator you get the answer 4.212 or the other one is negative 0.712 question number two solve the following quality equations by formula give your answers in three decimal places so this is the formula for you to find the x for the quality equations so first make sure the quadratic equation is written in general forms then determine the value of abc then you're going to substitute it into the formula given so start with questions a so the equation is x squared minus 4 x minus 7 equal to 0. therefore the a is equal to 1. the b is equal to negative 4 and c is equal to negative 7 so substitute it into the formula so negative b is negative bracket negative 4 plus minus square root so b squared is negative 4 squared minus 4 the a is 1 the c is negative 7 so square root this part then is going to be divided by 2a so a is 1. next this one gives you the answer of 4. solve this using your calculator the answer is 44 divide by 2. so the value of x one of it is four plus square roots 44 divided by two or the other one is four minus square root 44 divided by two so solve this using your calculator it will give you the answer of 5.317 or negative 1.317 for questions b so 2x squared plus 2x minus 1 equal to 0 in general form so the a is equal to 2 the b is equal to 2 as well c is equal to negative 1. so substitute this into the equation into the formula so negative the b is just 2 plus minus square root so b squared is 2 squared minus 4 the a is 2 the c is negative 1 so square root it over 2a next you want to have negative 2 plus minus so for the square root part after you solve it you have the answer of 12 divided by 4. so you have two value of x one of it is negative two plus square root of 12 divided by 4 or the other one is negative 2 minus negative square root 12 divided by 4. solve this using your calculator you have the answer of 0.366 or the other one is negative one point three six six questions c so three x square minus at x plus one equal to zero this one also in general form so the a is equal to 3 the b is equal to negative 8 and c is equal to positive 1 so substitute this into the formula so negative the b is a negative x plus minus square root so b squared is negative 8 squared minus 4 a is 3 c is 1. so square root this part over 2a which is 2 times 3. so here you're going to have a positive 8 plus minus so for the square root part after you solve it you want to have 52 divided by 6. so you have two value of x one of it is x plus square root 52 divided by 6 or the other one is at minus square root 52 divided by 6. solve this using a calculator the answer is 2.535 oh the other one is 0.131 questions t 4x squared minus 3x minus 2 equal to 0. so this quantity equations has been written in general form so we can solve this straight away so here the a is equal to 4 the b equal to negative 3 and c is equal to negative 2. so just substitute this into the formula so x is equal to negative b is negative 3 plus minus square root the b squared is negative 3 squared minus 4 the a is 4 the c is negative 2. so square root this part and divide by 2 a so here is going to be a positive 3 plus minus so the square root part you're going to have the answer of 41 here over 8. so the value of x 1 of it is 3 plus square root of 41 divided by and or the other one will be 3 minus square roots 41 divided by 8. solve these two using the calculator you will have the answer of one point one seven five or the other one is negative zero point four two five now move to questions e so bracket x minus 1 another one is x minus 3 equal to 5 so this one is not in general form so first we need to expand this part so after expanded you're going to have x square minus 3x minus x plus 3 equal to 5. so written all this in general form x squared so this part negative 3x minus x is minus 4x so plus 3 minus 5 equal to 0. so we can solve for the number here so this one is minus 2 equal to 0 then you're going to have it in general form so in this case the a is equal to 1 the b equal to negative 4 and c equal to negative 2. now replace it into the formula so x equal to negative b is negative 4 plus minus square root b squared is negative 4 squared minus 4 a c over 2a so here you want to have a positive 4 plus minus so the square root part is 24 over 2. so the value of x one of it is 4 plus square root of 24 divided by 2 and the other one is 4 minus square root 24 divided by 2. solve it using calculator you're going to have the answer of 4.449 or the other one is negative 0.449 question c equations f so bracket 2x minus 3 squared equal to 6 so this is the same as 2x minus 3 2x minus 3 equal to 6. so first expand this so you're going to have 4x squared minus 6x minus 6x plus 9 minus 6 equal to 0. so we're going to solve for the likes term so this one will give you the answer of negative 12 x plus 9 minus 6 is plus 3 equal to 0. so now you already have the equations in general form so a is equal to 4 b is equal to negative 12 and c is equal to positive 3. so substitute this into the formula so negative b bracket negative 12 plus minus b squared is negative 2 f squared minus 4 the a is 4 c is 3 over 2 a so here is going to be a positive 12 plus minus so the square root part the answer is 96 over 8. so the value of x1 of it is 12 plus square root 96 over 8 or the other one is 12 minus square root 96 over 8 so solve it using your calculator you're going to have the answer of 2.725 or the other one is 0.275 3a the length of the diagonal of a rectangle is 10 centimeter if the length is 2 centimeter longer than its width so the question is asking you to find the length and the width of the rectangle so i have a diagram of a rectangle here so the red line here represents the diagonals of the rectangle which has the measurements of 10 centimeter this will be the length and this is going to be the width let's say we represent the width with the letter x then for the length which is two centimeter longer it will be x plus two now this is a rectangle so here is going to be a 90 degree so we're going to look at the right angle triangle that we have here so we can solve this using the pythagoras so in pythagoras term is that a squared plus b squared equal to c squared the c is referred to the hypotenuse in this case which is the diagonal therefore the width is the a and the length is the b so a square plus b squared equal to c square when we represent it using the x and the x plus 2 is going to be x squared plus the b squared x plus 2 squared equal to 10 squared so expand this part you're going to have x square plus 4 x plus 4 equal to 100 so next we're going to solve for the likes term then and we're going to move this 100 to the left to create the general form of a quadratic equation so x squared plus x squared is 2x squared plus 4x plus 4 minus 100 equal to 0 4 minus 100 is minus 96 equal to 0 so we already have the quadratic equations in general form later on you can solve this using factorizations completing the square or by formula so let's say in this case i chose to solve it using formula therefore in the a here is equal to 2 the b is equal to 4 and c is equal to negative 96 so we want to substitute this into the formula so x is equal to negative b which is 4 plus minus square root b squared 4 squared minus 4 the a is 2 the c is negative 96 so we're going to square root this part over 2a which is 2 times 2 next going to have negative 4 plus minus so the square root part here if you solve it you're going to have the answer of say 184 over 2 times 2 which is 4. so the value of x one of it is going to be negative 4 plus square root 7 at 4 over 4 or the other one is going to be negative 4 minus square root 7 at 4 over 4. solve this using your calculator it will give you the answer 6 or negative 8. now i have two answer of x which one will be the correct one now compare it with the width that we have here the width is x and with representing the length therefore the x is equal to 6 because the length cannot be in negative forms therefore now we want to find the length the length is equal to x plus 2 now the x we know that is equal to 6. so i'm going to substitute the 6 into the x here therefore the length is 8 centimeter and for the width which is equal to x straight away the answer is 6 centimeter questions b find the measurements of a rectangle with a perimeter of 26 centimeter and an area of 40 centimeter square so let's say this is the rectangle so this is the length this is the width as we didn't know what's the length and width so let us represent the length as x and the width as y so this is going to be the x and y as well so to find the perimeter we're going to total up all of the sides in the rectangle so x plus y plus x plus y is equal to 26 therefore we will be able to form an equation of 2x plus 2y equal to 26 and for the area of a rectangle we're going to multiply the length with the width therefore x times y will equal to 40. now we have two variables in the equations here so we can solve this using either the eliminations method or the substitution methods in the simulstanes linear equate equations so we're going to label the first one here as equations number one this as equations number two so for this case i get to solve it using the substitutions method so from equations number two i'm going to let the x to be the subject of the formula therefore x is equal to 40 over y so let us label this as equations number three next we're going to substitute this equation number three into one which means we're going to substitute the x in our equations number one with 40 over y next we're going to have 2 bracket 40 over y plus 2 y equal to 26 so let us solve this first 2 times 40 you're going to have ft over y plus 2y equal to 26 next i'm going to get rid of the fractions here which is the denominator denominator y so i'm going to multiply each of the terms here with y so when at t over y times y the y here will be cancer so you're going to have ft plus 2y times y is 2y squared mine equal to 26y so this is a quadratic equation so next we're going to write it in general form so i'm going to move this 26y to the front 2y square minus 26y plus xt equal to 0. now to find the value of y you can either use the factorizations method completing the square or the formula so in this case i'm going to use the factorizations method so as i have the abc 2 negative 26 and 80 i can simplify it by dividing all of the term with number two so next i'm going to have y squared minus 13y plus 40 equal to zero so if you use the factorization method you want to have the bracket y minus x another bracket is y minus five equal to zero next y minus x equal to zero o y minus five equal to zero therefore y is equal to x or y is equal to five so if you use the method of completing the square or the formula you're going to get the same y here now we want to find what is the x so next we're going to substitute y equal to 8 or y equal to 5 into the equations number 3 to find the value of x so first when y is equal to x therefore we're going to substitute here x is equal to 40 over 8 which will give you the answer of 5. when y is equal to 5 so when we substitute it into uh the equations number 3 x is equal to 40 over 5 which is it now if you look at here we represent the length as x the width as y so the length must be longer than the width therefore the value of x must be greater than one so from here the first one when the y is equal to f the x is equal to five this one the five is less than the y but when the y equal to 5 the x will equal to x therefore the x is greater than the 5. so we can make a conclusion that the length the x is equal to x and the width which is the y is equal to 5. so the measurements the measurements of the rectangle so the measurement is equal to the length multiply the width therefore the measurement is at centimeter times five centimeter question four the diagram on the right should of trapezium picker is where pq is equal to x minus 1 ps equal to x plus 3 and qr equal to 3x plus 2 given the area of the trapezium is 17 centimeter squared finds the value of x so to find the area of the trapezoid we're going to use the formula 1 over 2 multiply a plus b times h so a b is referred to the parallel sides in the trapezoid so in this case the parallel sides is the ps and qr so ps we can label it as a and qr we can label it as b and h is the height of the trapezium in this case which is the pq so now we can form an equations using this formula so 1 over 2 multiply the a here is the x plus 3 then we're going to plus with the b which is 3x plus 2 then we're going to multiply it with the height in this case which is x minus 1. so this after you calculate it it will give you the answer of seventeen so next this is one over two we're going to move this one over two to the other side we're going to solve for the likes term for this bracket so x plus 3x is 4x plus 5 x minus 1 equal to 17 times 2 as we move 1 over 2 to the other side it will become times 2. so here we're going to do the expansions so you're going to have 4 x squared minus 4x plus 5x minus 5 equal to 34. so this is a quadratic equation so we want to write it in general form we're going to move everything to the left and solve for the likes term so 4x squared so negative 4x plus 5x is equal to positive 1 which is plus x minus 5 minus 34 equal to zero then we can solve for this one is equal to negative 39 equal to zero so next you already have the equations in general form you can solve it using either factorizations method completing the square or the formula so for these questions i can solve it using factorizations so you're going to have bracket x minus 3 another bracket is 4x plus 13 equal to 0. next x minus 3 equal to 0 or four x plus thirteen equal to zero so x is equal to three or x is equal to negative thirteen over four now i have two values of x which is 3 or negative 13 over 4. so which one is the correct one so for example if i substitute the x into the height here which is x minus 1 when x is equal to 3 3 minus 1 is 2 the answer is positive so when i substitute the x negative 13 over 4 here when we minus its minus 1 the answer for the height is negative 1 so the length must be in positive form therefore the correct answer for the x is x equal to 3.