K 7. Quadratic Equations. 1. Rewrite these polynomials in the form ax 2 + bx + c = 0. Identify the values of a, b and c:

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1 Qudrti Equtions The Null Ftor Lw Let's sy there re two numers nd. If # = then = or = (or oth re ) This mens tht if the produt of two epressions is zero, then t lest one of the epressions must e equl to zero. Here re some emples of how the null ftor lw is used to solve equtions: Solve for ^ + 3h= ^-1h^- h= ` # ^+ 3h= ` ^-1h# ^- h= ` = or + 3 = ^nullftor lwh ` - 1 = or - = ^nullftorlwh ` = or =-3 ` = 1 or = Ftorising to Solve Qudrti Equtions Type 1: + = Ftorise to 1 rket nd then use the null ftor lw to find two possile solutions. Solve for in these qudrti equtions + 4 = 3-15 = + 4 = 3-15 = ^ + 4h = ^3-15h = # ^+ 4h = # ^3-15h = ` = or + 4 = ` = or 3-15 = ` = or =-4 ` = or = 5 Type : + + = Ftorise to rkets nd then use the null ftor lw to find two possile solutions. Solve for in these qudrti equtions + = = = ` ^3-1h^+ 5h= ` ^- 3h^+ 4h= ` 3-1 = or + 5 = ` - 3 = or + 4 = ` = 3 or =-4 ` 3 = 1 or =-5 ` = 1 or = % Qudrti Equtions SERIES TOPIC Mthletis 1% 3P Lerning

2 Qudrti Equtions 1. Rewrite these polynomils in the form + + =. Identify the vlues of, nd : + 3+ = = - 7 = d ^ + 4h= e 3^4-5h= f ^+ 3h^- 7h= g ^3+ 5h^- 8h= h - 3^+ 4h^- 1h=. Solve these qudrti equtions for the missing vrile: ^ - h= ^ + 4h= ^+ 1h= d ^- 3h^+ 5h= e y^y- 3h= f ^+ 7h^3-8h= 1% Qudrti Equtions Mthletis 1% 3P Lerning SERIES TOPIC 5

3 Qudrti Equtions 3. Solve these qudrti equtions y ftorising: + 8 = = t - 6t = d 5-15 = e y -4y- 1 = f m -m- 3 = 6 1% Qudrti Equtions SERIES TOPIC Mthletis 1% 3P Lerning

4 Qudrti Equtions g n + 8n- 64 = h + 5 = 6 i 3 ^ + h= 15 j ^p+ 3h = 8p+ 7 1% Qudrti Equtions Mthletis 1% 3P Lerning SERIES TOPIC 7

5 Qudrti Equtions Completing the Squre The trinomil n't e ftorised into two rkets s esily s the emples in the previous setion. So to solve = the trinomil on the left needs to e trnsformed different wy. This is lled Completing the Squre is used in n emple to show how to omplete the squre elow: Complete the squre of Step 1: Write rket squred with plus hlf of the oeffiient of = Hlf = ^ + 3 h Sutrt Step : Sutrt the squre of the numer in the rket = ^ + 3 h - 9 Squred Step 3: Write in the onstnt term from the trinomil Step 4: Simplify ` = ^ + 3h - 8 = ^ + 3h = ^ + 3h - 8 In the ove emple the oeffiient of is 1. If it is not 1 then it must e ftored out first. Complete the squre of Step 1: Ftor out the oeffiient of using squre rkets = = 46-4 Hlf Step : Inside the squre rket write round rket squred with plus hlf the numer in front of in the squre rket Step 3: Inside the squre rket sutrt the squre of the numer in the round rket = 46 ^ - Sutrt = 46 ^ - h - 4@ Squred Step 4: Write the onstnt term in the squre rket from Step 1 ` = 4^ -h - 4 = 46^ -h -4 = 46^ -h - 6@ = 4^ -h % Qudrti Equtions SERIES TOPIC Mthletis 1% 3P Lerning

6 Qudrti Equtions 4. Complete the squre of the following trinomils: d Hint: 3 9 ` j = 4 e f Hint: ftorise -1 out using squre rkets 1% Qudrti Equtions Mthletis 1% 3P Lerning SERIES TOPIC 9

7 Qudrti Equtions Why is Completing the Squre Importnt? If trinomil in n eqution n't e ftorised esily into two rkets, then ompleting the squre n e used to solve the eqution. Solve for if = n't e ftorised esily into two rkets. Completing the squre n e used to show: = ^h -8 So the originl eqution n e rewritten s: ^ -3h - 8 = Simplify so tht the squred ket is y itself on the left hnd side: ^ -3h = + 8 ^ - 3h = 8 Solve for the rket y finding the squre root of oth sides: ^ - 3h = ^! 8h ` - 3 =! 8 Solve for the two possile solutions of : - 3 = 8 or - 3 =- 8 ` = 3+ 8 or = 3-8 Solutions re usully left in surd form, unless question speifies otherwise. Solve the eqution: = Using ompleting the squre: = 4( -) - 4 Rewrite the eqution in the ompleting the squre form: 4^ -h - 4 = Simplify so tht the squred rket is y itself on the left hnd side: 4^ - h = 4 ^ - h = 6 Solve for the rket y finding the squre root of oth sides: ^ - h = ^! 6h ` - =! 6 Solve for the two possile solutions of : - = 6 or - =- 6 ` = + 6 or = % Qudrti Equtions SERIES TOPIC Mthletis 1% 3P Lerning

8 Qudrti Equtions 5. Solve for in the following: ^ + h = 5 ^ - 3h = 16 3^ - 7h = 8 d - 5^ + 6h = Complete the squre nd then solve for in the following: = = 1% Qudrti Equtions Mthletis 1% 3P Lerning SERIES TOPIC 11

9 Qudrti Equtions 7. Complete the squre nd solve for the vrile in the following: q + 3q - = -m - m+ 5 = 4t + 8t - 1 = d = 1 1% Qudrti Equtions SERIES TOPIC Mthletis 1% 3P Lerning

10 Qudrti Equtions 8. Consider the eqution + + = where, nd re ny onstnts: Complete the squre for this eqution. Solve for. 1% Qudrti Equtions Mthletis 1% 3P Lerning SERIES TOPIC 13