Equations
Quadratic Equations 1
A quadratic equation has the form:
ax2 + bx + c = 0
The simplest method of solution is the Zero Product Rule.
If A x B = 0
then A = 0 or B = 0
Example 1
If (x-2)(x+3)=0
Example 2
Common Factor
then x-2=0 or x+3=0
x=2 or x = -3
(2x+1)(3x-2)=0
Most examples would need factorisation first.
If x2 -3x =0
then x(x-3)=0
gives x = 0 or x = 3
Remember to always remove the HCF.
Difference of two squares
If 4x2 - 25 =0
(2x-5)(2x+5) =0
Remember a2-b2=(a-b)(a+b)
Quadratic Factorisation
Make a positive in:
ax2 +bx +c = 0.
Hint:
If the last term is positive then the sign in each bracket is the same as the middle term.
Example 1
x2 -3x +2 =0
(x-2)(x-1)=0
x =2 or x = 1
Example 2
x2 + 5x + 6 = 0
(x+3)(x+2) =0
x = -3 or x = -2
Example 3
-x2 +2x -1 =0
now multiply by -1
x2 -2x +1 =0
(x-1)2 = 0
x = 1
Hint:
If the last term is negative then the sign in each bracket is different and the larger factor takes on the sign of the middle term.
x2 -2x -3 = 0
(x-3)(x+1)=0
x = 3 or x = -1
Example 5
x2 + 3x -18 =0
(x+6)(x-3)=0
x = -6 or x = 3