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Chapter 8: Problem 32

Use factoring and the zero product property to solve. $$u^{2}+9 u+18=0$$

Short Answer

Expert verified
The solutions are \(u = -6\) and \(u = -3\).

Step by step solution

01

- Identify and set up the quadratic equation

Start with the given quadratic equation: \[u^{2} + 9u + 18 = 0\]
02

- Factor the quadratic equation

Look for two numbers that multiply to 18 and add up to 9. These numbers are 6 and 3. Rewrite the equation as: \[u^2 + 6u + 3u + 18 = 0\] Then, factor by grouping: \[u(u + 6) + 3(u + 6) = 0\]
03

- Factor out the common binomial

Factor out the common binomial term \[(u + 6)(u + 3) = 0\]
04

- Apply the zero product property

Set each factor equal to zero and solve for \(u\): \[u + 6 = 0\] and \[u + 3 = 0\]
05

- Solve for \(u\)

Solve each equation: \[u = -6\] and \[u = -3\]

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Key Concepts

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

Factoring Quadratics
To solve a quadratic equation using factoring, you first need to break it down into simpler terms. Let's consider the equation given: \[u^{2} + 9u + 18 = 0\]

The goal is to express this equation as the product of two binomials. To do this, you need to find two numbers that multiply to the constant term (18) and add up to the linear coefficient (9). In this case, those numbers are 6 and 3 because \[6 \times 3 = 18\] and \[6 + 3 = 9\].

Therefore, we can rewrite the equation as: \[u^2 + 6u + 3u + 18 = 0\]
This makes it easier to then factor by grouping.
Zero Product Property
The Zero Product Property states that if \[a \times b = 0\], then either \[a = 0\] or \[b = 0\] (or both). This property is extremely useful when solving factored quadratic equations.

After factoring, you'll have a product of binomials equal to zero. From the example, we factored the equation to get: \[(u + 6)(u + 3) = 0\]

The Zero Product Property tells us that for this equation to hold true, at least one of the factors must be zero. So, we solve for each factor separately: \[u + 6 = 0\] and \[u + 3 = 0\]
This step narrows down the possible values of the variable.
Solving for Variables
Once you have used the Zero Product Property, the next step is to solve each equation for the variable. Each factor you've set to zero will give you a solution.

Consider the two equations: \[u + 6 = 0\] and \[u + 3 = 0\]
Solve for \[u\] in each case:
  • \u + 6 = 0\: \[u = -6\]
  • \u + 3 = 0\: \[u = -3\]

These are the solutions to the original quadratic equation.
Hence, the values of \[u\] that satisfy the equation \[u^{2} + 9u + 18 = 0\] are \[u = -6\] and \[u = -3\].

By following these steps鈥攆actoring the quadratic, applying the Zero Product Property, and solving the resulting equations鈥攜ou can solve any quadratic equation of this form.

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Most popular questions from this chapter

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