How do I factorise quadratic equations?
x2+x-6=y
Is the equation we will use to demonstrate how to factorise quadratics.
The first step involves using the basic shape of all quadratic factorisation:
ax2+bx+c=y
x2+x-6=y
(Cx+A)(Dx+B)=y
We must realize certain equalities that appear between the different expressions of this equation.
1.
Cx*Dx=ax2
C*Dx2=ax2
Cancelling x2
C*D=a
2.
A*B=c
3.
CxB+DxA=bx
BCx+ADx=bx
Cancelling x
BC+AD=b
This rigid layout can be used to factorise quadratics, but quadratics are all about pattern recognition and a small amount of practice goes a long way.
x2+x-6=y
ax2+bx+c=y
1. As our quadratic has no number multiplying on x^2 the first step of the solution is simple, we know that both C and D are equal to 1 as 1 only has one factor.
C*D=a
C*D=1
1*1=1
2. This is where paths in the solution diverge, as c in our equation, -6, has a number of factors
Those factors are:
+3*-2=-6
-3*+2=-6
+1*-6=-6
-1*+6=-6
A*B=-6
So we know the A and B are one of these factor pairs.
3. BC+AD=b
From step 1 in our solution, we know that both C and D are equal to 1. Meaning we can simplify our equation:
A+B=b
A+B=1
Now, from the factors we found in step 2, we must select a pair thats sum equals 1.
+3-2=1, so we know that A=+3 and B=-2 (it is arbritrary which number is assigned to each letter as the rest of the equation is the same).
ax2+bx+c=y
x2+x-6=y
(Cx+A)(Dx+B)=y
C=1
D=1
A=+3
B=-2
x2+x-6=y
(1x+3)(1x-2)=y
Finally, checking our answer:
1x*1x+3x-2x-6=y
x2+x-6=y
Following a rigid method is not recomended for solving quadratics, remember steps and the equalities that must occur, and practice, are the most important things.
