Where is the mistake?

Given that $p$ is a real number , then for what values of $p$ the equation ${(x+3p+1)}^{\frac{1}{3}} - x^ \frac{1}{3} = 1$ has real solutions.

My answer:

${(x+3p+1)}^{\frac{1}{3}} - x^ \frac{1}{3} - 1 = 0$ if $a+b+c = 0$ we have $a^3+b^3+c^3 = 3abc$ $x+ 3p + 1 -x -1 = 3[(x+3p+1)(x)]^\frac{1}{3}$ $3p = 3[(x+3p+1)(x)]^\frac{1}{3}$ $p^3 = x(x + 3p +1)$ $x^2 + (3p + 1)x -p^3 =0$ For real roots $D \geq 0$ $4p^3 + 9p^2 + 6p +1 \geq 0$ $(p+1)^2(4p+1) \geq 0$ therefore , $p \geq \frac{-1}{4} \cup -1.$

Please help me understand which step did I do wrong? And what is the correct solution for the above question?

#Algebra

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Comments

Now, note that

$a+b+c=0 \implies a^3 + b^3 + c^3 = 3abc$

But, $a^3 + b^3 + c^3 = 3abc \not \Rightarrow a+b+c=0$

So, $a+b+c=0$ is not equivalent to $a^3 + b^3 + c^3 = 3abc$ .

A better approach to solve this is putting $x=y^3 \iff x^{\frac{1}{3}} = y$ .

Then, the equation becomes:

$\begin{aligned}(y^3 + 3p +1)^{\frac{1}{3}} &= 1+y \\\iff y^3 + 3p +1 &= (1+y)^3 \\\iff y^2 + y -p &= 0 \end{aligned}$

So, for the roots of the original equation to be real, we need the discriminant of $y^2 + y - p=0$ to be non negative.

That is, $1+4p \geq 0 \\\iff p \geq -\frac{1}{4}$

A Former Brilliant Member - 4 years, 11 months ago

I got what you meant here but still i think the answer is wrong , check on wolframalpha putting p = -0.25 , it says no solution exists to the below equation for ${(x+0.25)}^{\frac{1}{3}} - x^ \frac{1}{3} - 1 = 0$

Ujjwal Mani Tripathi - 4 years, 11 months ago

Try $x = \frac{-1}{8}$

A Former Brilliant Member - 4 years, 11 months ago

@A Former Brilliant Member – Oh yes , i was working only with principal root value and hence was getting no solution

Ujjwal Mani Tripathi - 4 years, 11 months ago

WolframAlpha gives you the option to "use the real-valued root instead", (in blue just below the entry box), which in this case will give you a value of $x = -\frac{1}{8}$ as Deeparaj has pointed out.

Brian Charlesworth - 4 years, 11 months ago

What does $\frac{-1}{4} \cup 1$ mean?

Agnishom Chattopadhyay - 4 years, 11 months ago

i mean that p>= -0.25 and also p = -1 is a solutionof the inequality
$(p+1)^2(4p+1) \geq 0$

Ujjwal Mani Tripathi - 4 years, 11 months ago

@Brian Charlesworth @Jon Haussmann , your help will be valuable .

Ujjwal Mani Tripathi - 4 years, 11 months ago

I used the same approach as Deeparaj, but I think we need more than just the discriminant to be non-negative; in order to avoid complex values for $y = x^{\frac{1}{3}}$ we will require that $x \ge 0$ and thus also $y \ge 0$ . So we require that

$y = \dfrac{-1 + \sqrt{1 + 4p}}{2} \ge 0 \Longrightarrow \sqrt{1 + 4p} \ge 1 \Longrightarrow p \ge 0$ ,

which seems to be in agreement with results from WolframAlpha.

Brian Charlesworth - 4 years, 11 months ago

Don't we define $x^{\frac{1}{3}} = -(-x)^{\frac{1}{3}} \quad x < 0$ right?

A Former Brilliant Member - 4 years, 11 months ago

@A Former Brilliant Member – Yes, we could choose to work with the real-valued root, in which case WolframAlpha does return a value of $x = -\frac{1}{8}$ for $p = -\frac{1}{4}$ , and no solutions if $p \lt -\frac{1}{4}$ .

(WolframAlpha can be a bit difficult when taking fractional roots of negative numbers.)

Brian Charlesworth - 4 years, 11 months ago

@Brian Charlesworth – this is exactly what i was confused about . Thank you for helping me out with the wolframalpha query .

Ujjwal Mani Tripathi - 4 years, 11 months ago

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Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
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`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$