Problem 16 Solve. $$18 x+9 x^{2}=0$$... [FREE SOLUTION]

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Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 16

Solve. $$18 x+9 x^{2}=0$$

Short Answer

Expert verified

x = 0 or x = -2

Step by step solution

Factor the equation

Notice that both terms on the left side of the equation share a common factor of 9x. Factor out 9x from each term in the equation: \[9x(2 + x) = 0\]

Apply the zero-product property

The zero-product property states if the product of two factors is zero, at least one of the factors must be zero. Set each factor equal to zero:\[9x = 0\]\[2 + x = 0\]

Solve each equation

Solve for x in each equation:\[9x = 0\]Dividing both sides by 9 gives us:\[x = 0\]\[2 + x = 0\]Subtracting 2 from both sides gives us:\[x = -2\]

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Key Concepts

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

Factoring

Factoring is a useful technique for simplifying expressions and solving equations. It involves breaking down a complex expression into simpler components or factors that, when multiplied together, give back the original expression. In the given problem, the original equation was \[18x + 9x^2 = 0\]. To factor this equation, we first look for the greatest common factor (GCF) shared by all terms. Here, both terms share a common factor of 9x. By factoring 9x out of the expression, we get:\[9x(2 + x) = 0\].This step simplifies the problem and prepares it for the next concept: the zero-product property.

Zero-Product Property

The zero-product property is a fundamental principle used in algebra to solve equations involving products. It states that if the product of two factors is zero, then at least one of the factors must be zero. Mathematically, if \[a \times b = 0\], then either \[a = 0\] or \[b = 0\] (or both). Applying this property to our factored equation \[9x(2 + x) = 0\], we can set each factor equal to zero:

\[9x = 0\]
\[2 + x = 0\]

This gives us two simpler equations to solve, making the original problem much easier to handle.

Solving Equations

Solving equations involves finding the value or values of the variable that make the equation true. After factoring and applying the zero-product property, we are left with two simple equations to solve:\[9x = 0\] and \[2 + x = 0\]. We can solve \[9x = 0\] by dividing both sides by 9, which gives us \[x = 0\].Then, to solve \[2 + x = 0\], we subtract 2 from both sides to get \[x = -2\]. Thus, the solutions to the original equation are \[x = 0\] and \[x = -2\]. By following these steps, we've understood the process of factoring, using the zero-product property, and solving equations to find the solutions.

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91影视

Solve. $$18 x+9 x^{2}=0$$

Short Answer

Step by step solution

Factor the equation

Apply the zero-product property

Solve each equation

Key Concepts

Factoring

Zero-Product Property

Solving Equations

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