Help with reducing quadratic equation to final terms

**lilrhino** · June 7th 2007, 01:07 PM

$x^4 - 6x^2 + 8 = 0<br />$

$x^2=t$
$x^4=t^2$

a= 1, b = 6, c = 8

My answer using the quadratic formula:

$\Rightarrow x = \frac {6 \pm \sqrt {6^2 - 4(1)(8)}}{2(1)}$

$\Rightarrow x = \frac {6 \pm \sqrt {4}}{2}$

$\Rightarrow x = \frac {3 \pm \sqrt {2}}{}$

How do I reduce this further to get the final answer or is my answer correct? I always get confused. Thanks in advance.

**Jhevon** · June 7th 2007, 01:13 PM

Originally Posted by lilrhino

$x^4 - 6x^2 + 8 = 0<br />$

$x^2=t$
$x^4=t^2$

a= 1, b = 6, c = 8

My answer using the quadratic formula:

$\Rightarrow x = \frac {6 \pm \sqrt {6^2 - 4(1)(8)}}{2(1)}$

$\Rightarrow x = \frac {6 \pm \sqrt {4}}{2}$

$\Rightarrow x = \frac {3} \pm \frac { \sqrt {2}}{2}$

How do I reduce this further to get the final answer. I always get confused. Thanks in advance.

ok, so this is wrong, since b = -6 not 6. but we don't need the quadratic formula here

you replaced $x^2$ with $t$ , so you should have

$t^2 - 6t + 8 = 0$ .........now factor

$\Rightarrow (t - 4)(t - 2) = 0$

$\Rightarrow t = 4 \mbox { or } t = 2$

But $t = x^2$

$\Rightarrow x^2 = 4 \mbox { or } x^2 = 2$

$\Rightarrow x = \pm 2 \mbox { or } x = \pm \sqrt {2}$

**Soroban** · June 7th 2007, 01:17 PM

Hello, lilrhino!

You're making it unnecessarily complicated . . . and wrong.

$x^4 - 6x^2 + 8 \:= \:0$

Factor: . $(x^2 - 4)(x^2 - 2)\:=\:0$

And we have two equations to solve:

. . $x^2 - 4\:=\:0\quad\Rightarrow\quad x^2 \:=\:4\quad\Rightarrow\quad x \:=\: \pm2$

. . $x^2 - 2\:=\:0\quad\Rightarrow\quad x^2\:=\:2\quad\Rightarrow\quad x\:=\:\pm\sqrt{2}$

**lilrhino** · June 7th 2007, 01:19 PM

Originally Posted by Jhevon

ok, so this is wrong, since b = -6 not 6. but we don't need the quadratic formula here

you replaced $x^2$ with $t$ , so you should have

$t^2 - 6t + 8 = 0$ .........now factor

$\Rightarrow (t - 4)(t - 2) = 0$

$\Rightarrow t = 4 \mbox { or } t = 2$

But $t = x^2$

$\Rightarrow x^2 = 4 \mbox { or } x^2 = 2$

$\Rightarrow x = \pm 2 \mbox { or } x = \pm \sqrt {2}$

Thanks Jhevon, I have to watch my signs. I was looking at an example in the book, so I thought I had to use the quadratic formula even though it could be factored as you demonstrated. Thanks again!

**Jhevon** · June 7th 2007, 01:22 PM

Originally Posted by lilrhino

Thanks Jhevon, I have to watch my signs. I was looking at an example in the book, so I thought I had to use the quadratic formula even though it could be factored as you demonstrated. Thanks again!

yeah, the quadratic formula can be overkill at times. However, i do agree with Soroban that we are making it more complicated than it should be. Generally it helps students to visualize what to do by replacing a squared term with a single variable, but i believe that's wasting writing space. ordinarily, i'd just factor as Soroban did and cut out the middle man, but i didn't really want to shock you. you'd be surprised at some of the things students are shocked with

**lilrhino** · June 7th 2007, 01:30 PM

Originally Posted by Soroban

Hello, lilrhino!

You're making it unnecessarily complicated . . . and wrong.

Factor: . $(x^2 - 4)(x^2 - 2)\:=\:0$

And we have two equations to solve:

. . $x^2 - 4\:=\:0\quad\Rightarrow\quad x^2 \:=\:4\quad\Rightarrow\quad x \:=\: \pm2$

. . $x^2 - 2\:=\:0\quad\Rightarrow\quad x^2\:=\:2\quad\Rightarrow\quad x\:=\:\pm\sqrt{2}$

I do that often unfortunately. Math is not my strongest subject, so I sometimes make things more complicated that necessary. Thanks for your response.

**lilrhino** · June 7th 2007, 01:33 PM

Originally Posted by Jhevon

yeah, the quadratic formula can be overkill at times. However, i do agree with Soroban that we are making it more complicated than it should be. Generally it helps students to visualize what to do by replacing a squared term with a single variable, but i believe that's wasting writing space. ordinarily, i'd just factor as Soroban did and cut out the middle man, but i didn't really want to shock you. you'd be surprised at some of the things students are shocked with

I'm not easily shocked, I may spend more time thinking about it though

. It's easier the way Soroban demonstrated it actually, so I'm not shocked. Thanks...

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