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# Find the value of this without quadratic formula or completing the square

+2
226
7
+2216

Find the value of $$\frac{a^2}{a^4+a^2+1}$$ if $$\frac{a}{a^2+a+1}=\frac{1}{6}$$.

For some reason, I keep getting 1/36 every time I attempt it, and the answer is 1/24.

Please, someone give me a helping hint

May 3, 2019

#2
+105468
+2

The hint is that  $$(a^2+a+1)^2 \ne a^4+a^2+1$$

$$\frac{a}{a^2+a+1}=\frac{1}{6}\\ (\frac{a}{a^2+a+1})^2=\frac{1}{36}\\ (\frac{a}{a^2+a+1})^2=\frac{1}{36}\\ (\frac{a^2+a+1}{a})^2=36\\ \frac{a^4+2a^3+3a^2+2a+1}{a^2}=36\\ \frac{a^4+a^2+1+2a^3+2a^2+2a}{a^2}=36\\ \frac{(a^4+a^2+1)+2a(a^2+a+1)}{a^2}=36\\ \frac{(a^4+a^2+1)}{a^2}+\frac{2a(a^2+a+1)}{a^2}=36\\ \frac{(a^4+a^2+1)}{a^2}+2*\frac{(a^2+a+1)}{a}=36\\ \frac{(a^4+a^2+1)}{a^2}+12=36\\ \frac{(a^4+a^2+1)}{a^2}=24\\ \frac{a^2}{(a^4+a^2+1)}=\frac{1}{24}\\$$

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May 4, 2019

#1
+1

You can factor it as follows:
a^2 - 5 a + 1 = -1/4 (-2 a + sqrt(21) + 5) * (2 a + sqrt(21) - 5)
Split into 2 equations:
1/4 (-2 a + sqrt(21) + 5) =0
a =4.7912878....
(2 a + sqrt(21) - 5) =0
a =0.20871215.......
Sub any of the 2 values of a into the first part and you get:
= 1 / 24

May 3, 2019
edited by Guest  May 3, 2019
#2
+105468
+2

The hint is that  $$(a^2+a+1)^2 \ne a^4+a^2+1$$

$$\frac{a}{a^2+a+1}=\frac{1}{6}\\ (\frac{a}{a^2+a+1})^2=\frac{1}{36}\\ (\frac{a}{a^2+a+1})^2=\frac{1}{36}\\ (\frac{a^2+a+1}{a})^2=36\\ \frac{a^4+2a^3+3a^2+2a+1}{a^2}=36\\ \frac{a^4+a^2+1+2a^3+2a^2+2a}{a^2}=36\\ \frac{(a^4+a^2+1)+2a(a^2+a+1)}{a^2}=36\\ \frac{(a^4+a^2+1)}{a^2}+\frac{2a(a^2+a+1)}{a^2}=36\\ \frac{(a^4+a^2+1)}{a^2}+2*\frac{(a^2+a+1)}{a}=36\\ \frac{(a^4+a^2+1)}{a^2}+12=36\\ \frac{(a^4+a^2+1)}{a^2}=24\\ \frac{a^2}{(a^4+a^2+1)}=\frac{1}{24}\\$$

Melody May 4, 2019
#4
+104688
+1

Impressive, Melody   !!!!!

CPhill  May 4, 2019
#6
+105468
0

Thanks Chris

Melody  May 4, 2019
#3
+1

I also worked it out and I also got 1/24 so you are correct

May 4, 2019
#5
+105468
+1

Either that or we are both wrong

Melody  May 4, 2019
#7
+104688
+2

Nah....probably  < 1/24  probability that you are both wrong......

CPhill  May 4, 2019