Problem 51 - Elementary Algebra

Question

Solve. $$ 10 x 2+20 x=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = -2 \).

Step 1: Combine Like Terms

First, simplify the equation by combining like terms. Notice that both terms have a common factor which can be factored out from the equation. The equation is:\[ 10x^2 + 20x = 0 \] You can factor out the greatest common factor, which is 10x, from both terms.

Step 2: Factor the Equation

Factor out the greatest common factor, 10x, from the equation:\[ 10x(x + 2) = 0 \] This factored form tells us that if any part of the product is zero, the entire expression is zero.

Step 3: Apply Zero Product Property

According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero:1. \( 10x = 0 \)2. \( x + 2 = 0 \)

Step 4: Solve Each Factor

Solve each equation obtained from the Zero Product Property:1. For \( 10x = 0 \): Divide both sides by 10 to get: \[ x = 0 \]2. For \( x + 2 = 0 \): Subtract 2 from both sides to solve for x: \[ x = -2 \]

Key Concepts

Understanding Zero Product Property

The Zero Product Property is a key concept in solving quadratic equations like the one we have here. It states that if the product of two or more terms is zero, then at least one of the terms must be zero. This property is powerful when solving equations in factored form.

For example, consider the equation we worked with:

• Factored form: \(10x(x + 2) = 0\)

Using the zero product property, we can set each factor equal to zero to find possible values of \(x\):

• \(10x = 0\): Here, dividing both sides by 10 gives us \(x = 0\).
• \(x + 2 = 0\): Solving this gives us \(x = -2\).

This process shows that the solution to the original equation can be either \(x = 0\) or \(x = -2\) because both values will make the product \(10x(x + 2) = 0\). The zero product property helps break down complicated equations into manageable pieces.

Finding the Greatest Common Factor

Identifying the Greatest Common Factor (GCF) is a fundamental step in making quadratic equations easier to solve. It involves finding the largest factor that divides each term in an expression without leaving a remainder.

In our given equation,

• \(10x^2 + 20x\)

We notice both terms can be divided by 10. Moreover, both terms contain \(x\) as a factor.
Thus, the GCF here is \(10x\). By factoring out \(10x\), the equation simplifies to:

• \(10x(x + 2) = 0\)

This simplifies the equation, making it easier to apply further steps like the zero product property. Factoring with the greatest common factor is crucial because it simplifies expressions, revealing the equation's underlying structure and paving the way for easy problem-solving.

Steps to Solving Quadratic Equations

Solving quadratic equations efficiently requires a series of logical steps. Each step builds on the previous one, making the equation simpler to handle.

Here’s how you generally proceed:

• **Identify Terms**: Look for terms that can be combined or simplified.
• **Factor**: Use the greatest common factor to simplify the expression.
• **Apply Zero Product Property**: Set each factor to zero to find solutions.
• **Solve for Variables**: Solve the resulting simple equations to find the values for \(x\).

In our exercise, we:

• Factored \(10x^2 + 20x\) into \(10x(x + 2)\).
• Set each part equal to zero: \(10x = 0\) and \(x + 2 = 0\).
• Solved for \(x\) to get two solutions: \(x = 0\) and \(x = -2\).

These steps are repeatable for similar equations, consolidating a reliable method to solve quadratic equations across various problems. They frame a systematic approach ensuring success with comparable tasks.