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QUADRATICS ax 2 + bx + c

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MULTIPLYING BRACKETS (FOIL) x 2 +5x +6 Outside ( x+ 3)(x + 2) First x2x2 Inside Last +2x+3x +6

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MULTIPLYING BRACKETS (FOIL) (WITH MINUS NUMBERS) x 2 +2x -8 Outside ( x+ 4)(x - 2) First x2x2 Inside Last -2x+4x -8

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MULTIPLYING BRACKETS (FOIL) (WITH MINUS NUMBERS) x 2 -7x +12 Outside ( x-3)(x - 4) First x2x2 Inside Last -4x-3x+12

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y x y = x 2 +3x -4 When y =0 X = 1 or -4

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QUADRATICS You can find a quadratic from it’s roots

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QUADRATICS For example, x = 3 or -2 When y =0 (x - 3) =0 or (x + 2 ) =0 Multiply the brackets x 2 - x - 6 = 0

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FACTORISING AND SOLVING QUADRATICS x 2 +5x +6 ( x )(x ) The numbers have to add up to +5 and multiply to make The x’s have to multiply to make the first term

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CHECK IT OUT Outside ( x+ 3)(x + 2) First Inside Last x 2 +2x + 3x + 6 = (x 2 + 5x +6)

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FACTORISING AND SOLVING QUADRATICS 2x 2 - 4x - 6 ( 2x )(x ) The numbers have to add up to -4 and multiply to make The x’s have to multiply to make the first term 2x x -3 = - 6x -6x +2x = -4x

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EXAMPLES TO REMEMBER (a – b)(a + b) a 2 +ab –ab – b 2 = a 2 – b 2

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EXAMPLES TO REMEMBER The same applies to all these type of equations a 2 – 9 = (a-3)(a+3)

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EXAMPLES TO REMEMBER The same applies to all these type of equations 4a 2 – 36 = (2a -6)(2a+6)

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SOLVING USING THE QUADRATIC FORMULA ax 2 +bx +c is the standard form of a quadratic equation (where a, b and c represent numbers) to find x use the equation x = (-b ± √(b 2 – 4ac))/2a

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2x 2 +8x + 6 = 0 a = 2, b = 8, c = 6 x = (-8 ± √(8 2 – 4*2*6))/2*2 =(-8 ± √(64 – 48))/4 = ( -8 ± √16)/4 (-8 ± 4)/4 = -12/4 or -4/4 = -3 or -1 SOLVING USING THE QUADRATIC FORMULA

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x x + 34 = 0 x = (10 ± √(10 2 – 4*1*-34))/2 =(10 ± √(100 – 136))/2 = ( 10 ± √-36)/2 (10 ± 6i)/2 = 5 + 3i or 5 – 3i i is an imaginary number (i 2 = -1) Cannot have a zero square number so has to be multiplied by i 2 to make it positive

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