- #1
kenewbie
- 239
- 0
factoring "puzzle"
For what values of "a" can the following be factored and reduced:
[tex] \frac {a^2 - 2x -3}{x^2 - 4x + a} [/tex]
Ok, so the first thing on the list is to factor the top term.
[tex] x = \frac {-(-2) \pm \sqrt{(-2)^2 - 4 * 1 * (-3)} } {2 * 1} [/tex]
The roots are 3 & -1, and the factors then become
[tex] (x-3)(x+1) [/tex]
So, I want the denominator to become either both, or one of those factors.
I start off by rewriting the denominator as an equation
[tex] x = \frac {4 \pm \sqrt{16 - 4 * 1 * a} } {2} [/tex]
At this point I can tell that there are only solutions for [tex] a < 5 [/tex]
I know that I want the roots the polynomial to be either 3 or -1 (or both), so I substitute everything inside the square root with R, and check which values I need from the square root.
[tex]
\begin{align*}
\frac{4 + R}{2} = 3\\
4 + R = 6\\
R = 2\\
\end{align*}
[/tex]
and
[tex]
\begin{align*}
\frac{4 + R}{2} = -1\\
4 + R = -2\\
R = -6\\
\end{align*}
[/tex]
Since I would be hard pressed to get the square root back as negative 6, I conclude that I want the result of the square root to be 2.
In other words
[tex]
\begin{align*}
\sqrt{16 - 4 * 1 * a} = 2\\
16 - 4a = 4\\
a = 3\\
\end{align*}
[/tex]
Finally there. I factor the polynomial using [tex] a = 3 [/tex]
[tex] x = \frac {4 \pm \sqrt{16 - 4 * 3} } {2} [/tex]
The roots are 3 and 1 which gives the factors
[tex] (x-3)(x-1) [/tex]
And I can now factor the original rational expression
[tex]
\frac {a^2 - 2x -3}{x^2 - 4x + a} = \frac {(x-3)(x+1)} {(x-3)(x-1)} = \frac {x + 1}{ x -1}
[/tex]
However, there are two things that bug me to no end. First off the are two solutions listed, 3 and -5. I only found one of them, but I can't see where I went wrong and "missed" the second one?
The other thing is that I found my approach rather long winded, and I wondered if there was a better way to do this kind of problems?
Thanks for any feedback.
k
For what values of "a" can the following be factored and reduced:
[tex] \frac {a^2 - 2x -3}{x^2 - 4x + a} [/tex]
Ok, so the first thing on the list is to factor the top term.
[tex] x = \frac {-(-2) \pm \sqrt{(-2)^2 - 4 * 1 * (-3)} } {2 * 1} [/tex]
The roots are 3 & -1, and the factors then become
[tex] (x-3)(x+1) [/tex]
So, I want the denominator to become either both, or one of those factors.
I start off by rewriting the denominator as an equation
[tex] x = \frac {4 \pm \sqrt{16 - 4 * 1 * a} } {2} [/tex]
At this point I can tell that there are only solutions for [tex] a < 5 [/tex]
I know that I want the roots the polynomial to be either 3 or -1 (or both), so I substitute everything inside the square root with R, and check which values I need from the square root.
[tex]
\begin{align*}
\frac{4 + R}{2} = 3\\
4 + R = 6\\
R = 2\\
\end{align*}
[/tex]
and
[tex]
\begin{align*}
\frac{4 + R}{2} = -1\\
4 + R = -2\\
R = -6\\
\end{align*}
[/tex]
Since I would be hard pressed to get the square root back as negative 6, I conclude that I want the result of the square root to be 2.
In other words
[tex]
\begin{align*}
\sqrt{16 - 4 * 1 * a} = 2\\
16 - 4a = 4\\
a = 3\\
\end{align*}
[/tex]
Finally there. I factor the polynomial using [tex] a = 3 [/tex]
[tex] x = \frac {4 \pm \sqrt{16 - 4 * 3} } {2} [/tex]
The roots are 3 and 1 which gives the factors
[tex] (x-3)(x-1) [/tex]
And I can now factor the original rational expression
[tex]
\frac {a^2 - 2x -3}{x^2 - 4x + a} = \frac {(x-3)(x+1)} {(x-3)(x-1)} = \frac {x + 1}{ x -1}
[/tex]
However, there are two things that bug me to no end. First off the are two solutions listed, 3 and -5. I only found one of them, but I can't see where I went wrong and "missed" the second one?
The other thing is that I found my approach rather long winded, and I wondered if there was a better way to do this kind of problems?
Thanks for any feedback.
k