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Chapter 6: Problem 110

Solve using the principle of zero products. $$ 2 x-4 x^{2}=0 $$

Short Answer

Expert verified
The solutions are \( x = 0 \) and \( x = \frac{1}{2} \).

Step by step solution

01

- Factor the equation

First, rewrite the equation in standard form and factor it. The given equation is: \[ 2x - 4x^2 = 0 \]Factor out the common term, which is \(2x\):\[ 2x(1 - 2x) = 0 \]
02

- Apply the Zero Product Principle

The Zero Product Principle states that if the product of two factors is zero, at least one of the factors must be zero. Set each factor equal to zero:\[ 2x = 0 \]\[ 1 - 2x = 0 \]
03

- Solve for x

Solve each equation separately:For \(2x = 0\):\[ x = 0 \]For \(1 - 2x = 0\):\[ 1 = 2x \]\[ x = \frac{1}{2} \]
04

- Write the final solution

The solutions to the equation \(2x - 4x^2 = 0\) are:\[ x = 0 \] and \[ x = \frac{1}{2} \]

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Key Concepts

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

Factoring Equations
Factoring is a method used in algebra to simplify an equation by expressing it as a product of its factors. For instance, in the given exercise, we start with the equation \[ 2x - 4x^2 = 0 \]. To factor it, we look for common terms in each part of the equation. In this case, we see that both terms, \(2x\) and \(-4x^2\) have a common factor of \(2x\). By factoring out \(2x\), we can rewrite the equation as \[ 2x(1 - 2x) = 0 \]. Factoring makes the equation easier to solve because it breaks down complex expressions into simpler components.
Solving Quadratic Equations
Solving quadratic equations can often involve factoring the equation first. A quadratic equation is any equation that can be rewritten in the standard form \[ ax^2 + bx + c = 0 \]. In our example, the equation \[ 2x - 4x^2 = 0 \] is already set up for us to factor and solve. After factoring it to \[ 2x(1 - 2x) = 0 \], we utilize the Zero Product Property.This property states that if the product of two terms is zero, then at least one of these terms must be zero. Hence, we set each factor equal to zero: \[ 2x = 0 \] and \[ 1 - 2x = 0 \], and solve them individually. This process is crucial because it helps us find the values of \(x\) that satisfy the original equation.
Roots of Equations
The roots of an equation, also known as the solutions or zeros, are the values of the variable that satisfy the equation. In the exercise provided, after factoring the quadratic equation to \[ 2x(1 - 2x) = 0 \] and setting each factor to zero, we solve the resulting equations to find the roots. Solving \(2x = 0\) gives us \(x = 0\). For the equation \(1 - 2x = 0\), solving it results in \(x = \frac{1}{2}\). Therefore, the roots of the equation \(2x - 4x^2 = 0\) are \(x = 0\) and \(x = \frac{1}{2}\). These solutions represent the points where the original equation crosses the x-axis on a graph.

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Most popular questions from this chapter

To prepare for Section \(6.5,\) review the product and power rules for exponents and multiplication of polynomials (Sections 5.1 and 5.5 ). Simplify. [ 5.1] $$ \left(2 x^{2} y^{4}\right)^{3} $$

To prepare for Section \(6.5,\) review the product and power rules for exponents and multiplication of polynomials (Sections 5.1 and 5.5 ). Simplify. [ 5.1] $$ \left(-5 x^{2} y\right)^{3} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 16-a^{2}-2 a b-b^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ q^{2}+1 $$

Review multiplying binomials using FOIL $$ (a+4)(a-6) $$
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