75 a M1 2 M1 for squaring half of 5 H A1 A1 cao 2x2 + 10x 5 = 0 – Divide through by 2: b x2 + 5x = 0 – Completing the square: M1 M1 correctly simplifying equation (x + )2 = 0 – – M1 M1 completing the square (x + )2 = = A1 A1 cao Taking the square root of each side: x + = = ± 2.958 x x = 2.958 2.5 or = 2.5 – –2.958 – x x = 0.458 or = –5.458 Consider the 4 lines of his working. Line 1: he has forgotten the in the –5 A1 A1 for 2.958 (3 dp or more) equation. C1 C1 for showing the two possible solutions Line 2: he has squared the right-hand C1 C1 for two correct solutions (3 dp or more) side incorrectly. Line 3: the should be and the should be c B1 B1 for line 1 comments Line 4: there should be two solutions. B1 B1 for line 2 comments + 5x 5 = 0 x2 – (x + )2 5 = ()2 the in – Don’t forget –5 the original equation. B1 B1 for line 3 comments (x + )2 5 = Square the whole of the – bracket on the right-hand side. (x + )2 = 5 + Remember, when you add B1 B1 for line 4 comments a term on one side, you must also add d it on the other side. (x + ) = ± = ± Be careful when taking square roots of fractions. that when you Don’t forget find square root there is a positive and a negative root. C2 C1 for commenting on each line C1 for clear positive comments that would be deemed helpful 14
75 a b c d 2x2 + 10x – 5 = 0 Divide through by 2: x2 + 5x – = 0 Completing the square: (x + )2 – – = 0 (x + )2 = = Taking the square root of each side: x + = = ± 2.958 x = 2.958 – 2.5 or x = –2.958 – 2.5 x = 0.458 or x = –5.458 Consider the 4 lines of his working. Line 1: he has forgotten the –5 in the equation. Line 2: he has squared the right-hand side incorrectly. Line 3: the should be and the should be Line 4: there should be two solutions. x2 + 5x – 5 = 0 (x + )2 – 5 = ()2 Don’t forget the –5 in the original equation. (x + )2 – 5 = Square the whole of the bracket on the right-hand side. (x + )2 = 5 + Remember, when you add a term on one side, you must also add it on the other side. (x + ) = ± = ± Be careful when taking square roots of fractions. Don’t forget that when you find square root there is a positive and a negative root. M1 A1 M1 M1 A1 A1 C1 C1 B1 B1 B1 B1 C2 2 M1 for squaring half of 5 A1 cao M1 correctly simplifying equation M1 completing the square A1 cao A1 for 2.958 (3 dp or more) C1 for showing the two possible solutions C1 for two correct solutions (3 dp or more) B1 for line 1 comments B1 for line 2 comments B1 for line 3 comments B1 for line 4 comments C1 for commenting on each line C1 for clear positive comments that would be deemed helpful H 14