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Chapter 5: Problem 22

Solve each equation. $$ 3 x^{2}-27 x=0 $$

Short Answer

Expert verified
The solutions are x = 0 and x = 9.

Step by step solution

01

- Factor Out the Common Term

The given equation is \[3x^2 - 27x = 0\].First, identify the common term in both parts of the equation. In this case, both terms have a factor of 3 and x. Factor out 3x:\[3x(x - 9) = 0\].
02

- Apply the Zero Product Property

According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible equations to solve:\[3x = 0\]\[x - 9 = 0\]
03

- Solve Each Equation

Now, solve each equation separately:1. For the equation \[3x = 0\]:\[x = 0\]2. For the equation \[x - 9 = 0\]:\[x = 9\]

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Key Concepts

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

Factoring Quadratic Expressions
Factoring quadratic expressions is a very important skill in solving quadratic equations. When you factor a quadratic expression, you are breaking it down into simpler pieces called factors. These factors, when multiplied together, give you the original quadratic expression.

For example, in the given equation \[3x^2 - 27x = 0\], both terms share a common factor of 3 and x. By factoring out the common term, we re-write it as \[3x(x - 9) = 0\]. This makes it easier to solve the equation by setting each factor to zero, which leads us to the next concept: zero product property.
Zero Product Property
The Zero Product Property is an essential principle in algebra. It states that if the product of two or more factors is zero, then at least one of the factors must be zero.

In simpler terms, if you have an equation of the form \[A \times B = 0\], then it must be true that either \[A = 0\] or \[B = 0\] (or both). This concept helps us break down complex equations into simpler, solvable parts.

Applying it to our example, \[3x(x - 9) = 0\], we apply the Zero Product Property and get two possible equations: \[3x = 0\] and \[x - 9 = 0\]. Each of these equations can then be solved separately, leading us to the final solutions.
Solving Linear Equations
Once you've factored the quadratic expression and applied the Zero Product Property, you'll often end up with simpler linear equations to solve. Linear equations are much easier to handle. They have a standard format of \[ax + b = 0\].

Let's look at the equations we derived from our example:
  • \[3x = 0\]
  • \[x - 9 = 0\]

The first equation, \[3x = 0\], can be solved by dividing both sides by 3, resulting in \[x = 0\].

The second equation, \[x - 9 = 0\], can be solved by adding 9 to both sides, yielding \[x = 9\].

Thus, the solutions to the original quadratic equation \[3x^2 - 27x = 0\] are \[x = 0\] and \[x = 9\].

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