Factoring a Quadratic Expression

[aricho]

sorry this is such a stupid question...but i have been sitting her for half an hour trying to solve it.

what does

a^2-4a-12

factorise to?

 

[LeonhardEuler]

If you're ever absolutely stuck on a factorization problem, you can use the quadratic equation to find the roots. Then, suppose the roots are a and b. The equation will factor to (x-a)(x-b). In this case, that might not be necessary. Think: You need two numbers that multiply to give you -12 and add to give you -4. List the pairs of integers that you could multiply to get -12. Which of these add to give -4?

 

[aricho]

its -6 and 2 isn't it?

gosh I'm a fool sometimes

 

[LeonhardEuler]

That's it.

 

[aricho]

arrrrrrgggghhhhhh

how about x^2-6x-9

 

[LeonhardEuler]

In this case the answer comes out irrational. Make sure you copied it right. Otherwise use the quadratic formula.

 

[HallsofIvy]

If you are every really, really stuck on a (quadratic) factoring problem you can do as Leonhard Euler said: solve the equation using the quadratic formula. If the roots are a and b, then the quadratic factors as (x-a)(x-b).

For example, to solve a²-4a-12= 0, a=
$$\frac{4\pm\sqrt{16-4(1)(-12)}}{2}= \frac{4\pm\sqrt{64}}{2}= \frac{4\pm8}{2}$$ so that a= (4+8)/2= 6 and a= (4-8)/2= -2. The factors are (a-6)(a-(-2))= (a-6)(a+2).

More importantly, x²-6x-9 does not have rational factors (are you sure this wasn't x²- 6x+ 9?) but can be factored if we solve the equation
x²- 6x+ 9= 0.

I would be inclined to "complete the square": x²- 6x= 9 and we complete the square on the left by adding (-6/2)²= (-3)²= 9 (no surprize there!). x²- 6x+ 9= (x- 3)²= 9+ 9= 18= 2*9.
$$x-3= \pm\3\sqrt{2}$$
so the solutions are $$x= 3- 3\sqrt{2}$$ and $$x= 3+ 3\sqrt{2}$$
$$x^2-6x-9= (x-3+3\sqrt{2})(x-3-3\sqrt{2})$$.

You could also get that factorization by completing the square originally:
$$x^2- 6x- 9= x^2- 6x+ (9-9)- 9= x^2- 6x+ 9- 18= (x-3)^2- 18$$
Now factor as the difference of two squares:
$$x^2-6x-9= (x-3)^2-18= ((x-3)+\sqrt{18})((x-3)-\sqrt{18})$$
$$= (x-3+3\sqrt{2})(x-3-3\sqrt{3})$$

 

[sr6622]

I'm glad no one did the problem for him...