Factorising Quadratics:
No constant:
eg: 15x² + 3x Only 1 bracket Look for common factors = 3x(5x+1) |
"One x squared"
eg: x² + 5x - 24 (sum of) ? + ? = (b) = 5 (product of) ? x ? = (c) = -24 Gives 8 and -3 = (x+8)(x-3) |
"More than one x squared"
eg: 2x² + 7x + 6 (sum of) ? + ? = 7 (product of) ? x ? = (ac) 2 x 6 = 12 Gives 4 and 3 Split the x terms into the two given numbers 2x² + 4x + 3x + 6 Factorise the first two terms and the last two terms 2x(x+2) : 3(x+2) Take out the common factor of (x+2) = (2x+3)(x+2) |
Examples:
Factorise: 3x² + 10x + 7
b = 10 ac = 21 = 7 and 3 3x² + 7x + 3x + 7 x(3x+7) : 1(3x+7) (3x+7)(x+1) Factorise: 5x² + 7x - 12 b = 7 ac = -60 = 20 and 3 5x² + 20x - 3x -12 5x(x+4) : -3(x+4) (5x-3)(x+4) |
Factorise: 2x² + 5xy + 3y²
b = 5xy ac = 6(x²)(y²) = 3xy and 2xy 2x² = 3xy + 2xy + 3y² x(2x+3y) : y(2x+3y) (2x+3y)(x+y) Factorise: 8x² - 2xy - 3y² b = -2xy ac = -24(x²)(y²) = -6xy and 4xy 8x² - 6xy + 4xy - 3y² 2x(4x-3y) : y(4x-3y) (4x-3y)(2x+y) |
Solving Equations:
In order to solve any equation, they MUST EQUAL ZERO!
No Constant:
eg: 15x² + 3x = 0 Factorise: 3x(5x+1) = 0 : 3x x (5x+1) = 0 3x = 0 (/3) 5x+1 = 0 (-1) x = 0 5x = -1 (/5) x = -1/5 x = 0 or x = -1/5 |
"One or more x squared"
eg:x² + 5x - 24 = 0 Factorise: (x+8)(x-3) = 0 x+8 = 0 (-8) x-3 = 0 (+3) x = -8 x = 3 x = -8 or x = 3 |
Difference of two squares (DoTs)
eg: x² = 25 Make the equation equal "0" (-25) x²-25 = 0 Factorise: (x+5)(x-5) = 0 x+5 = 0 (-5) x-5 = 0 (+5) x = -5 x = 5 x = -5 or x = 5 |