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Chapter 3: Problem 42

Find all real numbers \(x\) that satisfy the indicated equation. $$ x-\sqrt{x}=12 $$

Short Answer

Expert verified
The real number $x$ that satisfies the equation \(x - \sqrt{x} = 12\) is approximately \(x \approx 30.912\).

Step by step solution

01

Isolate the square root term

Add \(\sqrt{x}\) to both sides of the equation: \[ x - \sqrt{x} + \sqrt{x} = 12 + \sqrt{x} \] This simplifies to: \[ x = 12 + \sqrt{x} \]
02

Square both sides of the equation

In order to eliminate the square root, we need to square both sides of the equation: \[ (x)^2 = (12 + \sqrt{x})^2 \] Expand and simplify: \[ x^2 = 144 + 24\sqrt{x} + x \]
03

Bring all terms to one side of the equation

To make it a quadratic equation, subtract x and 144 from both sides of the equation: \[ x^2 - x - 144 = 24\sqrt{x} \]
04

Square both sides of the equation again

We still have a square root in the equation, so we need to square both sides again: \[ (x^2 - x - 144)^2 = (24\sqrt{x})^2 \] Simplify: \[ x^4 - 2x^3 + 289x^2 - 576x + 20736 = 576x \]
05

Bring all terms to one side of the equation and simplify

Subtract 576x from both sides to obtain a polynomial equation: \[ x^4 - 2x^3 + 289x^2 - 1152x + 20736 = 0 \]
06

Solve the polynomial equation

As this is a degree 4 polynomial equation, it's not easy to solve it analytically. However, we can observe that x=0 is an extraneous root, since it doesn't satisfy the original equation. Thus, we can use synthetic division or polynomial long division to divide the equation by x, then solve the resulting cubic polynomial equation numerically or graphically to find the real roots. In conclusion, to find all real numbers x that satisfy the given equation, we need to follow these steps: isolate the square root term, square both sides of the equation twice, bring all terms to one side of the equation and simplify, and finally solve the polynomial equation using appropriate techniques.

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