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Chapter 9: Problem 27

Solve. $$9 x^{2}-3 x=0$$

Short Answer

Expert verified
The solution set is S = {0, 1/3}

Step by step solution

01

Factor out common factors

First, factor out the common factors. In this case, you can factor out \(x\)and 3, so the equation becomes \(3x(3x-1) = 0\)
02

Set each factor equal to zero and solve.

Next, set each factor equal to zero and solve for \(x\). You get two possible solutions. If \(3x = 0\), then \(x = 0\). If \(3x - 1 = 0\), solving for \(x\) gives \(x = 1/3\)
03

Write the solution set.

Present the set of solutions as S = {0, 1/3}. That is, the solutions are \(x = 0\) and \(x = 1/3\)

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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 an essential skill for solving certain types of equations, especially when it comes to quadratics like the given equation: \[9x^2 - 3x = 0.\] Breaking it down into a product of simpler expressions can make it easier to find the values of \(x\). When you factor, look for common factors in each term. Here, the terms are \(9x^2\) and \(-3x\). Both share a factor of \(3x\), so you can factor this out:
  • Divide each term by \(3x\).
  • This yields \(3x(3x - 1)\).
This method simplifies the quadratic into a product of two factors: \(3x\) and \(3x - 1\). Recognizing common factors and understanding how to factor them efficiently is crucial, as it sets the stage for finding the solutions to the quadratic equation.
Solution Set
After factoring the quadratic expression, the next step is determining the solution set, or the values of \(x\) that satisfy the equation. This involves setting each of the individual factors equal to zero.
Start with:
  • \(3x = 0\): Solving for \(x\), divide both sides by 3 to get \(x = 0\).
  • \(3x - 1 = 0\): Solve by adding 1 to both sides and then dividing by 3, resulting in \(x = \frac{1}{3}\).
Once these calculations are made, the solutions \(x = 0\) and \(x = \frac{1}{3}\) form the solution set.
Presenting the solution set as \(S = \{0, \frac{1}{3}\}\) encapsulates all possible values for \(x\) that satisfy the original equation. Expressing solutions clearly in set notation is a helpful practice when communicating the results.
Roots of Equations
The roots of equations, particularly in the context of quadratic equations, refer to the values of \(x\) that make the equation true. For the quadratic equation \(9x^2 - 3x = 0\), the roots are found once the quadratic expression is factored and each factor is set equal to zero. These roots, \(x = 0\) and \(x = \frac{1}{3}\), are the solutions where the quadratic touches or crosses the x-axis on a graph.
Understanding that roots are the points at which the expression equals zero is key:
  • They are critical in determining the number of times an equation crosses the x-axis.
  • They help define the function's behavior graphically.
Grasping the connection between roots and the graph of a quadratic equation reinforces the importance of solving for these values.

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Most popular questions from this chapter

Use the quadratic formula to find all real solutions of each equation. SEE EXAMPLE 3. (OBJECTIVE 1) $$x^{2}+3 x+1=0$$

Simplify. Write each result in a + bi form. $$(5+\sqrt{-3})(2-\sqrt{-3})$$

Set up an equation to solve. SEE EXAMPLE 5. (OBJECTIVE 3) One plane heads west from an airport flying at \(200 \mathrm{mph}\). One hour later, a second plane heads north from the same airport, flying at the same speed. When will the planes be 1,000 miles apart?

Use the quadratic formula to determine whether there are real number solutions to each equation. SEE EXAMPLE 4. (OBJECTIVE 2) $$x^{2}+2 x+7=0$$

The binomial \(b^{2}-4 a c\) is called the discriminant. From its value, you can predict whether the solutions of a given quadratic equation are real or nonreal numbers. Explain.
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