Problem 26 - Beginning and Intermediate Algebra

Question

Solve each equation. $$10 a^{2}+20 a=0$$

Step-by-Step Solution

Answer

The simplified equation is \(10(a^2 + 2a) = 0\), and using the zero-product property, we find the solutions: \(a = 0\) and \(a = -2\).

Step 1: Factor out the Greatest Common Factor (GCF)

In the equation \(10a^2 + 20a = 0\), the greatest common factor for both terms is 10. We'll factor this out of the equation: \[10(a^2 + 2a) = 0\]

Step 2: Apply the Zero-Product Property

Since the product of two numbers is equal to zero, one of the numbers must be zero. This is called the zero-product property. So we have two possibilities to check: \[10 = 0\] or \[a^2 + 2a = 0\] We know that 10 is never equal to zero, so our second equation will give us the possible values of "a".

Step 3: Simplify and Solve for "a"

From the equation \(a^2 + 2a = 0\), we can factor out the common factor of "a": \[a(a + 2) = 0\] Now, applying the zero-product property again: \[a = 0\] or \[a + 2 = 0\] For the second equation, we have: \[a = -2\] So the solutions for the given quadratic equation are \(a = 0\) and \(a = -2\).

Key Concepts

Factoring

Factoring is a method used to simplify algebraic expressions. It involves breaking down a complex equation or expression into simpler parts, called factors. Think of factoring like unpacking a box to see what's inside; each part helps you understand the whole better. In the case of quadratics, factoring is crucial because it transforms a seemingly complicated polynomial into two or more manageable terms.

The equation from our exercise, \(10a^2 + 20a = 0\), is a perfect example. We can make it simpler by taking out the greatest common element, which is 10 in this instance. This step is essential because it reduces the polynomial to a form that's easier to handle, resulting in \(10(a^2 + 2a) = 0\).

• Factoring allows us to inspect the polynomial more closely.
• It sets the stage for applying the zero-product property.
• It simplifies solving for "a" by breaking the problem into smaller parts.

By breaking down equations in this way, factoring turns what might seem like a complex equation into something clear and manageable.

Zero-Product Property

The zero-product property is a fundamental concept in solving quadratic equations. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. This is incredibly useful, especially in the context of factored equations.

In our example, after factoring \(10(a^2 + 2a) = 0\), we use the zero-product property. The idea is that, for the equation to hold true, the factor \(a^2 + 2a\) itself must be zero, since 10 cannot be zero. This step simplifies the solving process dramatically.

• Ensures that we find valid solutions like \(a = 0\) and \(a = -2\).
• Makes complex equations manageable by breaking them into singular components.
• Highlights the essence of each factor in relation to zero.

The zero-product property is like a puzzle piece that, when put in the right spot, reveals key solutions to the problem.

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest number that divides all terms in the polynomial without leaving a remainder. Finding the GCF is the first step in the factoring process and is particularly useful for simplifying quadratic equations.

In the problem \(10a^2 + 20a = 0\), the GCF is 10. This common factor is factored out, yielding \(10(a^2 + 2a) = 0\). Why is this important? Because factoring out the GCF makes the equation easier to solve and is a typical first step when working with quadratic equations.

• Identifying the GCF helps to reduce coefficients, leading to simpler expressions.
• Makes subsequent steps like applying the zero-product property more straightforward.
• Serves as a critical tool in equation solving and mathematical simplification.

Always check for a GCF first; it can save you time and effort in solving quadratic equations.