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Chapter 21: Chapter 21

Problem 627

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Find all solutions of the equation $\mathrm{x}^{3}-3 \mathrm{x}^{2}-10 \mathrm{x}=0$

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The solutions for the equation \(x^3 - 3x^2 -10x = 0\) are \(x = 0\), \(x = 5\), and \(x = -2\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Factor out the common term

We can factor out the common term (x) from the given equation. This gives us: \[x(x^2 - 3x - 10) = 0\]

Step 2: Factor the quadratic equation

Now, we need to factor the quadratic equation \(x^2 - 3x - 10\). To do that, we find two numbers whose product is equal to the constant term (-10) and whose sum is equal to the coefficient of the x term (-3). We find that these two numbers are -5 and 2. So, we can write the quadratic equation as: \[(x - 5)(x + 2) = 0\] Now, our factored equation looks like this: \[x(x - 5)(x + 2) = 0\]

Step 3: Find the solutions (roots)

The solutions (roots) of the equation are the values of x that make the equation equal to zero. Based on the factored form, we can see that there are three possible solutions: 1. \(x = 0\): The equation is equal to zero when x is 0. 2. \(x - 5 = 0\): The equation is equal to zero when x is 5. 3. \(x + 2 = 0\): The equation is equal to zero when x is -2. Therefore, the solutions for the equation \(x^3 - 3x^2 -10x = 0\) are \(x = 0\), \(x = 5\), and \(x = -2\).

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