/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Find all real values of \(x\) su... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 44

Find all real values of \(x\) such that \(f(x)=0\). $$f(x)=x^{3}-x^{2}-4 x+4$$

Short Answer

Expert verified
The real solutions to the function \(f(x) = 0\) are \(x = 1, x = 2, x = -2\).

Step by step solution

01

Write Down The Given Function

The given function is \( f(x) = x^{3} - x^{2} - 4x + 4 \). The task is to find the real roots of this equation, i.e., to find \(x\) for which \(f(x) = 0\). Hence, we write the equation as \( 0 = x^{3} - x^{2} - 4x + 4 \).
02

Factorize the Cubic Equation

The next step is to factorize the cubic equation. This can be quite tricky for cubic equations, but here it can be factored as \( (x-1) * (x^{2} - 4) = 0 \). In fact, \(x^{2} - 4\) can be further factored as it is a difference of squares, resulting in \( (x-1) * (x - 2) * (x + 2) = 0 \)
03

Solve The Equation

We proceed by setting each factor equal to zero, as this is a basic property of equations (if a * b = 0, then either a = 0 or b = 0). This results in three solutions: \[ x - 1 = 0 \Rightarrow x = 1 \] \[ x - 2 = 0 \Rightarrow x = 2 \] \[ x + 2 = 0 \Rightarrow x = -2 \] So we have three solutions: \(x = 1, x = 2, x = -2\).

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