• Jul 31st 2007, 02:52 PM
jimmic
Hi, I am trying to teach myself algebra from a book. It was going fairly smooth and I understand most of what I have read and practiced. However, I came to a section on quadratic equations whereupon I was presented with a basic form x^2 = 3. It was simple enough to understand that the answer must be either x = 1.73 or -1.73 without any experience with quadratic equations.

Now, the very next paragraph I was presented with this (x - 1)^2 - 4 = 0 and the simple solution that x = 3 or -1. Then my problem begins as the author goes on to say that the equation (x - 1)^2 - 4 can be "simplified" to x^2 - 2x - 3. How is this possible? where does he get 2x? I have checked the numbers and they are ofcourse correct but how has he made this assumption without knowing the value of x? Is there some sort of mathematical trick I have missed out on here?

My level of Math is far from impressive so any advice would be helpful please. I would normally just read or re-read to try to understand but one minute the author is making sense then it's asif he just skips a page and assumes you understand :)
• Jul 31st 2007, 03:07 PM
ThePerfectHacker
Quote:

Originally Posted by jimmic
Hi, I am trying to teach myself algebra from a book. It was going fairly smooth and I understand most of what I have read and practiced. However, I came to a section on quadratic equations whereupon I was presented with a basic form x^2 = 3. It was simple enough to understand that the answer must be either x = 1.73 or -1.73 without any experience with quadratic equations.

Now, the very next paragraph I was presented with this (x - 1)^2 - 4 = 0 and the simple solution that x = 3 or -1. Then my problem begins as the author goes on to say that the equation (x - 1)^2 - 4 can be "simplified" to x^2 - 2x - 3. How is this possible? where does he get 2x? I have checked the numbers and they are ofcourse correct but how has he made this assumption without knowing the value of x? Is there some sort of mathematical trick I have missed out on here?

My level of Math is far from impressive so any advice would be helpful please. I would normally just read or re-read to try to understand but one minute the author is making sense then it's asif he just skips a page and assumes you understand :)

(x-1)^2=(x-1)(x-1)

Now expand this.
• Jul 31st 2007, 03:16 PM
topsquark
Quote:

Originally Posted by jimmic
Now, the very next paragraph I was presented with this (x - 1)^2 - 4 = 0 and the simple solution that x = 3 or -1. Then my problem begins as the author goes on to say that the equation (x - 1)^2 - 4 can be "simplified" to x^2 - 2x - 3. How is this possible? where does he get 2x? I have checked the numbers and they are ofcourse correct but how has he made this assumption without knowing the value of x? Is there some sort of mathematical trick I have missed out on here?

Do you know the FOIL method?

As TPH said:
$(x - 1)^2 = (x - 1)(x - 1) = x \cdot x + x \cdot (-1) + (-1) \cdot x + (-1) \cdot (-1) = x^2 - x -x + 1 = x^2 -2x + 1$

-Dan
• Jul 31st 2007, 03:46 PM
Krizalid
Quote:

Originally Posted by jimmic
Now, the very next paragraph I was presented with this (x - 1)^2 - 4 = 0 and the simple solution that x = 3 or -1. Then my problem begins as the author goes on to say that the equation (x - 1)^2 - 4 can be "simplified" to x^2 - 2x - 3. How is this possible?

We can use the difference of perfect squares as follows


\begin{aligned}
\left( {x - 1} \right)^2 - 4 &= 0\\
\left[ {\left( {x - 1} \right) + 2} \right]\left[ {\left( {x - 1} \right) - 2} \right] &= 0\\
\left( {x + 1} \right)\left( {x - 3} \right) &= 0\\
x^2 - 2x - 3 &= 0
\end{aligned}

$\emph{As desired}~\blacksquare$

P.S.: actually, the last step isn't necessary to get the roots of the equation.