MHB Can X^3 Equal sqrt{81} with MathMagic Lite?

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The equation x^3 = sqrt{81} has three roots, including two complex conjugates that are often overlooked. The correct approach involves factoring the equation as a difference of cubes. While one solution is real, the other two are complex, and all must be considered unless specified otherwise. MathMagic Lite is favored for its ease of use, although LaTeX is preferred for its superior presentation of mathematical expressions. Overall, the discussion emphasizes the importance of recognizing all roots in polynomial equations.
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Is the following correct?
 

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RTCNTC said:
Is the following correct?

yes
 
You're actually missing two roots of the equation:

$$x^3=\sqrt{81}$$

Write as difference of cubes:

$$x^3-\left(3^{\frac{2}{3}}\right)^3=0$$

Factor:

$$\left(x-3^{\frac{2}{3}}\right)\left(x^2+3^{\frac{2}{3}}x+3^{\frac{4}{3}}\right)=0$$

The quadratic factor will yield the two complex conjugate roots. ;)
 
MarkFL said:
You're actually missing two roots of the equation:

$$x^3=\sqrt{81}$$

Write as difference of cubes:

$$x^3-\left(3^{\frac{2}{3}}\right)^3=0$$

Factor:

$$\left(x-3^{\frac{2}{3}}\right)\left(x^2+3^{\frac{2}{3}}x+3^{\frac{4}{3}}\right)=0$$

The quadratic factor will yield the two complex conjugate roots. ;)

What do you think of MathMagic Lite? Cool, right?

1. Are you saying that my answer is wrong?

2. How badly was your area hit by Irma?
 
RTCNTC said:
What do you think of MathMagic Lite? Cool, right?

I much prefer $\LaTeX$...it looks a lot better, can be quoted, and seen in the "Topic Review" element. You can fix the last issue by attaching the images inline, but $\LaTeX$ is still the far superior solution for posting math expressions, IMHO. :)

RTCNTC said:
1. Are you saying that my answer is wrong?

No, I'm just saying there are two other values of $x$ that satisfy the given equation. Now, the answer you gave is real, and the other two solutions are complex, so unless the problem states only to look for real solutions, then we must give all 3.

RTCNTC said:
2. How badly was your area hit by Irma?

It was hit pretty hard, but not as hard as pretty much all points to the south of me. :)
 
I will continue using MathMagic Lite. It is easy and less stressful to use.
 
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