Quadratics ax2 + bx + c

Multiplying brackets (FOIL) x2 +2x +3x +6 x2 +5x +6 Outside ( x+ 3)(x + 2) Inside Last First

Multiplying brackets (FOIL) (with minus numbers) x2 -2x +4x -8 x2 +2x -8 Outside ( x+ 4)(x - 2) Inside Last First

Multiplying brackets (FOIL) (with minus numbers) x2 -4x -3x +12 x2 -7x +12 Outside ( x-3)(x - 4) Inside Last First

y y = x2 +3x -4 When y =0 X = 1 or -4 x

You can find a quadratic from it’s roots Quadratics You can find a quadratic from it’s roots

Quadratics For example, x = 3 or -2 When y =0 (x - 3) =0 or (x + 2 ) =0 Multiply the brackets x2 - x - 6 = 0

Factorising and solving quadratics The x’s have to multiply to make the first term x2 +5x +6 ( x )(x ) +3 +2 The numbers have to add up to +5 and multiply to make +6

Check it out Outside ( x+ 3)(x + 2) Last Inside First x2 +2x + 3x + 6 = (x2 + 5x +6)

Factorising and solving quadratics The x’s have to multiply to make the first term 2x2 - 4x - 6 2x x -3 = -6x -6x +2x = -4x ( 2x )(x ) +2 - 3 The numbers have to add up to -4 and multiply to make -6

Examples to remember (a – b)(a + b) a2 +ab –ab – b2 = a2 – b2

The same applies to all these type of equations Examples to remember The same applies to all these type of equations a2 – 9 = (a-3)(a+3)

The same applies to all these type of equations Examples to remember The same applies to all these type of equations 4a2 – 36 = (2a -6)(2a+6)

Solving using the Quadratic formula ax2 +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 ± √(b2 – 4ac))/2a

Solving using the Quadratic formula 2x2 +8x + 6 = 0 a = 2, b = 8 , c = 6 x = (-8 ± √(82 – 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 x2 -10x + 34 = 0 x = (10 ± √(102 – 4*1*-34))/2 =(10 ± √(100 – 136))/2 = ( 10 ± √-36)/2 (10 ± 6i)/2 = 5 + 3i or 5 – 3i i is an imaginary number (i2 = -1) Cannot have a zero square number so has to be multiplied by i2 to make it positive