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Chapter 2: Problem 62

Find all real numbers \(x\) that satisfy the indicated equation. $$ x^{2 / 3}+3 x^{1 / 3}=10 $$

Short Answer

Expert verified
The real numbers \(x\) that satisfy the equation \(x^{2/3} + 3x^{1/3} = 10\) are \(x = -125\) and \(x = 8\).

Step by step solution

01

Substitute a new variable

Let's substitute a new variable \(u = x^{1/3}\). This way we can rewrite the equation using the new variable as: \(u^2 + 3u = 10\)
02

Rearrange into a quadratic equation

Now that we have the equation in terms of the new variable, we can rearrange it into a standard quadratic equation. Subtract 10 from both sides to get: \(u^2 + 3u - 10 = 0\)
03

Solve the quadratic equation

Now we need to solve the quadratic equation to find the possible values of u. We can factor the equation as follows: \((u + 5)(u - 2) = 0\) From this, we can see that there are two possible values for u: \(u_1 = -5\) or \(u_2 = 2\)
04

Reverse the substitution

Now we need to reverse the substitution to find the actual values of x. Recall that we made the substitution \(u = x^{1/3}\). Therefore, we have: \(x_1^{1/3} = -5\) and \(x_2^{1/3} = 2\) Now, we need to find the cube of both sides of both equations: \(x_1 = (-5)^3 = -125\) and \(x_2 = (2)^3 = 8\)
05

Check the solutions in the original equation

Finally, we need to ensure that the solutions for x we found indeed satisfy the original equation \(x^{2/3} + 3x^{1/3} = 10\). Let's check: For \(x_1 = -125\): \((-125)^{2/3} + 3(-125)^{1/3} = 25 - 15 = 10\) For \(x_2 = 8\): \((8)^{2/3} + 3(8)^{1/3} = 4 + 6 = 10\) Both solutions satisfy the original equation. Therefore, the real numbers x that satisfy the indicated equation are \(x = -125\) and \(x = 8\).

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