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Solve An Equation

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anemone

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Feb 14, 2012
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Solve in real numbers the following equation:

\(\displaystyle (7x+1)^{\frac{1}{3}}+(-x^2+x+8)^{\frac{1}{3}}+(x^2-8x-1)^{\frac{1}{3}}=2.\)
 

MarkFL

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Feb 24, 2012
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I see by inspection that $x=0,1,9$ are solutions. (Smirk)
 
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anemone

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Feb 14, 2012
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MarkFL

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Feb 24, 2012
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Cool! But there are a total of 4 solutions to this problem...
Sweet! Then I didn't hog them all, and have left one for someone else to eyeball! (Happy)
 

MarkFL

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Feb 24, 2012
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I have "eyeballed" that $x=-1$ is a solution as well. (Smoking)

Time to be serious though...(Nod)

I would first arrange the equation as:

\(\displaystyle (7x+1)^{\frac{1}{3}}+\left(x^2-8x-1 \right)^{\frac{1}{3}}=2+\left(x^2-x-8 \right)^{\frac{1}{3}}\)

Let:

\(\displaystyle u=7x+1\)

\(\displaystyle v=x^2-8x-1\)

\(\displaystyle w=x^2-x-8\)

and we have:

(1) \(\displaystyle u^{\frac{1}{3}}+v^{\frac{1}{3}}=2+w^{\frac{1}{3}}\)

Cubing both sides of the equation, we find after simplification:

\(\displaystyle (uv)^{\frac{1}{3}}\left(u^{\frac{1}{3}}+v^{\frac{1}{3}} \right)=2w^{\frac{1}{3}}\left(2+w^{\frac{1}{3}} \right)\)

Using (1), we may write:

\(\displaystyle (uv)^{\frac{1}{3}}\left(u^{\frac{1}{3}}+v^{\frac{1}{3}} \right)=2w^{\frac{1}{3}}\left(u^{\frac{1}{3}}+v^{ \frac{1}{3}} \right)\)

We may arrange this as:

\(\displaystyle \left(u^{\frac{1}{3}}+v^{\frac{1}{3}} \right)\left((uv)^{\frac{1}{3}}-2w^{\frac{1}{3}} \right)=0\)

Using the zero-factor property, this yields the cases:

i) \(\displaystyle u^{\frac{1}{3}}+v^{\frac{1}{3}}=0\)

\(\displaystyle u=-v\)

Back-substituting for $u$ and $v$, we find:

\(\displaystyle 7x+1=-x^2+8x+1\)

\(\displaystyle x^2-x=0\)

\(\displaystyle x(x-1)=0\)

\(\displaystyle x=0,1\)

ii) \(\displaystyle (uv)^{\frac{1}{3}}-2w^{\frac{1}{3}}=0\)

\(\displaystyle uv=8w\)

Back substituting for $u$, $v$, and $w$, we find:

\(\displaystyle (7x+1)\left(x^2-8x-1 \right)=8\left(x^2-x-8 \right)\)

After simplification, we obtain:

\(\displaystyle (x+1)(x-1)(x-9)=0\)

\(\displaystyle x=-1,1,9\)

we have already ascertained that the four solutions we found are valid, hence we may state the real solutions are:

\(\displaystyle x=-1,0,1,9\)
 
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anemone

MHB POTW Director
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Feb 14, 2012
3,681
Time to be serious though...(Nod)

I would first arrange the equation as:

\(\displaystyle (7x+1)^{\frac{1}{3}}+\left(x^2-8x-1 \right)^{\frac{1}{3}}=2+\left(x^2-x-8 \right)^{\frac{1}{3}}\)

Let:

\(\displaystyle u=7x+1\)

\(\displaystyle v=x^2-8x-1\)

\(\displaystyle w=x^2-x-8\)

and we have:

(1) \(\displaystyle u^{\frac{1}{3}}+v^{\frac{1}{3}}=2+w^{\frac{1}{3}}\)

Cubing both sides of the equation, we find after simplification:

\(\displaystyle (uv)^{\frac{1}{3}}\left(u^{\frac{1}{3}}+v^{\frac{1}{3}} \right)=2w^{\frac{1}{3}}\left(2+w^{\frac{1}{3}} \right)\)

Using (1), we may write:

\(\displaystyle (uv)^{\frac{1}{3}}\left(u^{\frac{1}{3}}+v^{\frac{1}{3}} \right)=2w^{\frac{1}{3}}\left(u^{\frac{1}{3}}+v^{ \frac{1}{3}} \right)\)

We may arrange this as:

\(\displaystyle \left(u^{\frac{1}{3}}+v^{\frac{1}{3}} \right)\left((uv)^{\frac{1}{3}}-2w^{\frac{1}{3}} \right)=0\)

Using the zero-factor property, this yields the cases:

i) \(\displaystyle u^{\frac{1}{3}}+v^{\frac{1}{3}}=0\)

\(\displaystyle u=-v\)

Back-substituting for $u$ and $v$, we find:

\(\displaystyle 7x+1=-x^2+8x+1\)

\(\displaystyle x^2-x=0\)

\(\displaystyle x(x-1)=0\)

\(\displaystyle x=0,1\)

ii) \(\displaystyle (uv)^{\frac{1}{3}}-2w^{\frac{1}{3}}=0\)

\(\displaystyle uv=8w\)

Back substituting for $u$, $v$, and $w$, we find:

\(\displaystyle (7x+1)\left(x^2-8x-1 \right)=8\left(x^2-x-8 \right)\)

After simplification, we obtain:

\(\displaystyle (x+1)(x-1)(x-9)=0\)

\(\displaystyle x=-1,1,9\)

we have already ascertained that the four solutions we found are valid, hence we may state the real solutions are:

\(\displaystyle x=-1,0,1,9\)
Well done, MarkFL!

But...

I have "eyeballed" that $x=-1$ is a solution as well. (Smoking)
...smoking is not a good habit and is not allowed in my thread, hehehe...:p(Smile)
 

MarkFL

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Feb 24, 2012
13,775
It's an E cigarette...(Smirk)...so no worries. (Happy)