This is the answer in the book but I don't know how they got from
x^2-5root(2)X+12=0
to
(x-3root(2))(x-2root(2))
Help please?
Thank you!
well, you can use the quadratic formula to find the roots of the equation, which the roots are
[-b+sqrt(b^2-4ac)]/2a and [-b-sqrt(b^2-4ac)]/2a
where a=1, b=-5root(2), c=12
after you get the roots, rewrite your equation in the form of
(x-root1)(x-root2)=0
First note that $\displaystyle (x+x_0)(x+x_1)= x^2+ (x_0+ x_1)x+ x_0x_1$. That means that in order to factor a polynomial of the form $\displaystyle x^2+ ax+ b$ you look for numbers $\displaystyle x_0$ and $\displaystyle x_1$ such that $\displaystyle x_0+ x_1= a$ and $\displaystyle x_0x_1= b$. In this particular example $\displaystyle a= -5\sqrt{2}$ and $\displaystyle b= 12$ so you need to solve $\displaystyle x_0+ x_1= -5\sqrt{2}$ and $\displaystyle x_0x_1= 12$. Since that product does NOT have a $\displaystyle \sqrt{2}$ in it, it is clear that both $\displaystyle x_0$ and $\displaystyle x_1$ must have a factor of $\displaystyle \sqrt{2}$.
So, now, we are thinking $\displaystyle x_0= m\sqrt{2}$ and $\displaystyle x_1= n\sqrt{2}$ for some numbers m and n. $\displaystyle x_0+ x_1= (m+n)\sqrt{2}= -5\sqrt{2}$ and $\displaystyle x_0x_1= 2mn= 12$ so we now have m+ n= 5 and mn= 6. -2 and -3 satisfy those equations so we can take $\displaystyle x_0= -2\sqrt{2}$ and $\displaystyle x_1= -3\sqrt{2}$ and have $\displaystyle x^2- 5\sqrt{2}x+ 12= (x- 2\sqrt{2})(x- 3\sqrt{2})$