# (Help) Simplifying by factoring question for 45x^6 = 9x^4

#### Tofi

##### New member
Hi, so I've been self-teaching myself maths as a hobby for a little while now. And I came across this problem that's confused me a bit. I thought I followed all the necessary steps to simplify it, but my answer book told me I didn't take it that one step further. I'm not sure how it got this final answer and my book frustratingly hasn't taught me how it came to this conclusion.

$$\displaystyle 45x^6\, =\, 9x^4$$

$$\displaystyle \dfrac{45x^6}{\color{blue}{x^4}}\, =\, \dfrac{9x^4}{\color{blue}{x^4}}$$

$$\displaystyle 45x^2\, =\, 9$$

$$\displaystyle \dfrac{45x^2}{\color{blue}{45}}\, =\, \dfrac{9}{\color{blue}{45}}$$

$$\displaystyle x^2\, =\, \dfrac{1}{5}$$

$$\displaystyle \color{blue}{\sqrt{\strut \color{black}{x^2}\,}}\, =\, \color{blue}{\sqrt{\strut \color{black}{\frac{1}{5}}\,}}$$

$$\displaystyle x\, =\, \color{blue}{\pm}\dfrac{1}{\sqrt{\strut 5\,}}$$

$$\displaystyle \color{red}{ x\, =\, \pm\, \dfrac{\sqrt{\strut 5\,}}{5} }$$

Thanks in advance! (Black and blue is my working out. Red is what I missed according to the answers page)

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#### mmm4444bot

##### Super Moderator
Staff member
Hi. Your result and the red result are two different forms of the same answer.

Were you instructed to always "rationalize the denominator", by chance? :cool:

To rationalize a denominator means to multiply both the numerator and denominator by the same radical expression, to get a ratio with no radical in the denominator.

For example, rationalize the denominator in 5·√(3/7)

$$\displaystyle \dfrac{5\sqrt{3}}{\sqrt{7}} \; \cdot \;$$$$\displaystyle \dfrac{\sqrt{7}}{\sqrt{7}}$$$$\displaystyle = \dfrac{5\sqrt{21}}{7}$$

The denominator has been rationalized.

#### lookagain

##### Senior Member
(Black and blue is my working out. Red is what I missed according to the answers page)

The answers page is incorrect if you presented it fully. In the context of this specific problem,
it is not allowable to divide by the variable, because information is lost.

$$\displaystyle 45x^6 = 9x^4$$

$$\displaystyle 45x^6 - 9x^4 = 0$$

$$\displaystyle 9x^4(5x^2 - 1) = 0$$

From the quartic factor set equal to zero, that gives x = 0.

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#### Tofi

##### New member
Hi. Your result and the red result are two different forms of the same answer.

Were you instructed to always "rationalize the denominator", by chance? :cool:

To rationalize a denominator means to multiply both the numerator and denominator by the same radical expression, to get a ratio with no radical in the denominator.

For example, rationalize the denominator in 5·√(3/7)

$$\displaystyle \dfrac{5\sqrt{3}}{\sqrt{7}} \; \cdot \;$$$$\displaystyle \dfrac{\sqrt{7}}{\sqrt{7}}$$$$\displaystyle = \dfrac{5\sqrt{21}}{7}$$

The denominator has been rationalized.
Thanks!