Finding Horizontal Intercepts

**maca404** · January 18th 2013, 11:34 PM

I have the function $x^4-2*x^2-1$

I have fount the dirivative
$4x^3-4x$

and all stationary points as the first part of the question asked but I am stuck on the second half it says :

(b) By using the substitution $u = x^2$ , find the horizontal intercepts.

For the most part I dont really understand the question and have no idea were to start to solve it, Any ideas ?

**MarkFL** · January 19th 2013, 12:21 AM

I'm assuming by horizontal intercepts, they mean roots. Using the suggested substitution, we may write:

$f(u)=u^2-2u-1=0$

Now, use the quadratic formula (or complete the square) to find the roots in terms of $u$ , then back substitute for $u$ to find the 4 roots in terms of $x$ .

**maca404** · January 19th 2013, 02:26 AM

Yes its is the roots they are looking for and from graphing I know thay are -1.554 and 1.554, I can solve for the roots o the equation you suggested I just cant see how I am going to get back to the correct answer ?

**Prove It** · January 19th 2013, 03:33 PM

OK, first of all, do NOT round off until the very end. Keep answers exact where possible.

Now, as you were instructed by Mark, if you have let $\displaystyle \begin{align*} u = x^2 \end{align*}$ then you end up with $\displaystyle \begin{align*} u^2 - 2u - 1 = 0 \end{align*}$ . Solve this EXACTLY using the Quadratic Formula or Completing the Square. Then you will have the possible values for $\displaystyle \begin{align*} u \end{align*}$ .

Once you have the values for $\displaystyle \begin{align*} u \end{align*}$ , remember that $\displaystyle \begin{align*} u = x^2 \end{align*}$ and use this to find the values of $\displaystyle \begin{align*} x \end{align*}$ .

**maca404** · January 19th 2013, 08:41 PM

Ok using the quadratic equation I come up with

U= - $0.414213562$ and U = $2.414213562$

so if $U = X^2$ then $X=sqrt(u)$

and indeed $Sqrt(2.414213562)=1.554$

but I cant take the square of the neg so how do I get the value for -1.554 ?

**Prove It** · January 20th 2013, 07:23 PM

Originally Posted by maca404

Ok using the quadratic equation I come up with

U= - $0.414213562$ and U = $2.414213562$

so if $U = X^2$ then $X=sqrt(u)$

and indeed $Sqrt(2.414213562)=1.554$

but I cant take the square of the neg so how do I get the value for -1.554 ?

Do NOT round off your answers. What are your answers for u in their EXACT surd form?

**maca404** · January 21st 2013, 12:23 AM

You mean in the form x=1+sqrt(2) and x=1-sqrt*(2)

**Prove It** · January 21st 2013, 04:28 PM

Originally Posted by maca404

You mean in the form x=1+sqrt(2) and x=1-sqrt*(2)

Almost. Actually it's $\displaystyle \begin{align*} u = 1 \pm \sqrt{2} \end{align*}$ , not x.

Now since you know that $\displaystyle \begin{align*} u = x^2 \end{align*}$ , that gives $\displaystyle \begin{align*} x^2 = 1 \pm \sqrt{2} \end{align*}$ . How will you solve for x?

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