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Chapter 6: Problem 130

What is wrong with solving \(x^{2}=3 x\) by dividing both sides of the equation by \(x ?\)

Short Answer

Expert verified
By dividing by \[ x \], the solution \[ x = 0 \] is missed because it assumes \[ x eq 0 \].

Step by step solution

01

Understand the given equation

The equation is given as \[ x^2 = 3x \].
02

Simplify by factoring

Before dividing by any variable, factor the equation. \[ x^2 - 3x = 0 \] can be factored as \[ x(x - 3) = 0 \].
03

Apply the Zero Product Property

According to the Zero Product Property, if \[ ab = 0 \], then either \[ a = 0 \] or \[ b = 0 \]. Here, it implies \[ x = 0 \] or \[ x - 3 = 0 \].
04

Solve for x

From the previous step, the solutions are \[ x = 0 \] and \[ x = 3 \].
05

Identify the mistake in dividing by x

If you divide both sides of \[ x^2 = 3x \] by \[ x \], you have \[ x = 3 \]. This method ignores the solution \[ x = 0 \] because dividing by \[ x \] assumes \[ x eq 0 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring
Factoring is an essential technique for solving quadratic equations. The main goal is to express the equation as a product of simpler expressions. This makes solving more manageable.

Take the equation \[ x^2 - 3x = 0 \]. To factor it, we look for common terms. Here, both terms share an \[ x \]. So, we factor out the \[ x \], resulting in \[ x(x - 3) = 0 \].

We use factoring because it sets the stage for the Zero Product Property. Without factoring, solving quadratic equations becomes challenging and sometimes impossible using basic algebra.
zero product property
The Zero Product Property helps us solve equations that have been factored. If a product of two numbers is zero, at least one of the numbers must be zero.

For example, in our factored equation \[ x(x - 3) = 0 \], one of the factors, either \[ x \] or \[ x - 3 \], must be zero for the product to be zero.

So, we set each factor equal to zero:
  • \[ x = 0 \]
  • \[ x - 3 = 0 \], which simplifies to \[ x = 3 \]
This gives us the solutions \[ x = 0 \] and \[ x = 3 \].
division by variable
Dividing by a variable in an equation can be risky and may lead to incorrect solutions. Here's why.

Consider the equation \[ x^2 = 3x \]. If you divide both sides by \[ x \], you get \[ x = 3 \]. While this is a solution, it omits another possible solution, \[ x = 0 \].

This happens because division by \[ x \] assumes \[ x \] is not zero. However, \[ x \] could be zero, which is why we miss that solution.

By factoring instead, we ensure all solutions are considered: \[ x(x - 3) = 0 \] allows us to find both \[ x = 0 \] and \[ x = 3 \].

Avoid dividing by a variable directly. Use factoring to keep all potential solutions on the table.

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Most popular questions from this chapter

Find the zeros of each function. $$ f(x)=x^{3}+x^{2}-4 x-4 $$

Factor completely. Assume that variables in exponents represent positive integers. $$ x^{8}-2^{8} $$

Factor completely. Assume that variables in exponents represent positive integers. $$ 9 x^{2 n}-6 x^{n}+1 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 9 x^{4}-25 x^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ a^{3}-a b^{2}-2 a^{2}+2 b^{2} $$
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