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Chapter 11: Problem 49

Complete each factoring. See Section 5.1. $$ -2 x^{2}+6 x=-\left(x^{2}-3 x\right) $$

Short Answer

Expert verified
x = 0 or x = 3

Step by step solution

01

Simplify the Equation

First, simplify the given equation by distributing the negative sign on the right-hand side: -2x^{2} + 6x = -(x^{2} - 3x) -2x^{2} + 6x = -x^{2} + 3x
02

Rearrange All Terms to One Side

Move all terms to one side of the equation to set it equal to zero: -2x^{2} + 6x + x^{2} - 3x = 0 Simplify the resulting equation: -x^{2} + 3x = 0
03

Factor Out the Greatest Common Factor (GCF)

Factor out the greatest common factor, which is x: x(-x + 3) = 0
04

Apply the Zero Product Property

Now apply the zero product property. Set each factor equal to zero and solve for x: x = 0 or -x + 3 = 0 Solve the second equation for x: -x + 3 = 0 x = 3

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Key Concepts

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

Greatest Common Factor
The greatest common factor, or GCF, is the largest factor that can divide all the terms in an expression without leaving a remainder. To find the GCF, identify the common factors in the terms and choose the highest number that divides all terms. For instance, in the expression -2x^2 + 6x + x^2 - 3x = 0, we observe terms like -2x^2, 6x, -x^2, and -3x. The variable 'x' is present in all terms. Thus, 'x' is the GCF. Factoring the GCF simplifies the equation, making it easier to solve.
Zero Product Property
The zero product property is a fundamental principle in algebra which states that if the product of two factors is zero, then at least one of the factors must be zero. This property is key to solving quadratic equations. In our example, after factoring out the GCF, we get {x(-x + 3) = 0}. Using the zero product property, we set each factor equal to zero: {x = 0} and {-x + 3 = 0}. Solving these simple equations gives us the solutions {x = 0} and {x = 3}. Applying this property helps us break down the problem into simpler parts.
Solving Quadratic Equations
Solving quadratic equations usually involves getting the equation in standard form {ax^2 + bx + c = 0} and then finding the roots of the equation. There are multiple ways to solve them, including factoring, using the quadratic formula, or completing the square. In our example, the equation was simplified to {-x^2 + 3x = 0}. Factoring gave us {x(-x + 3) = 0}. Using the zero product property, we solved for {x = 0} and {x = 3}. These solutions represent the points where the graph of the quadratic equation intersects the x-axis. Breaking down each step helps approach these equations methodically.

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