Problem 3 - College Algebra with Corequisite Support

Question

When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?

Step-by-Step Solution

Answer

To use the zero product property for factoring and solving.

Step 1: Understanding the Goal

The goal of solving a quadratic equation by factoring is to find its roots. The roots are the values of the variable that make the equation true.

Step 2: Writing the Equation in Standard Form

Before we can factor a quadratic equation, we first need to write it in standard form. The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.

Step 3: Setting the Equation to Zero

To solve the quadratic equation by factoring, we move all terms to one side of the equation to set it equal to 0. This is essential because setting the equation to zero allows us to use the zero product property.

Step 4: Applying the Zero Product Property

Once the equation is factored and set to zero, we can apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This leads us to find the values of the variable that satisfy this condition.

Key Concepts

Factoring Method

The factoring method is a systematic approach to solving quadratic equations by turning a complicated expression into a product of simpler expressions, or factors. When solving quadratics by factoring, you essentially reverse the distributive property used in multiplication.

Here's how it works:

• First, ensure the equation is in standard form, meaning all terms are on one side with a zero on the opposite side.
• Next, identify or determine the factors of the equation. The goal is to find two or more expressions that multiply together to give the original quadratic equation.
• Common methods used in factoring include the "difference of squares", "perfect square trinomials", and "factor by grouping".

The primary advantage of the factoring method is its simplicity and effectiveness for specific quadratics. However, not all quadratic equations can be easily factored, especially if they're complex or don't have nice integer factors. In such cases, other methods like completing the square or the quadratic formula might be more appropriate.

Standard Form

The standard form of a quadratic equation is written as \( ax^2 + bx + c = 0 \). This form is essential because it lays the groundwork for solving quadratics, especially when using the factoring method.

The components of the equation are:

• \( ax^2 \): The quadratic term, where \( a \) is the coefficient of \( x^2 \) and should not be zero.
• \( bx \): The linear term, with \( b \) being its coefficient.
• \( c \): The constant term, which can be any real number.

Writing the equation in standard form involves collecting all terms on one side, typically the left, and ordering them by decreasing power of \( x \). Ensuring the equation equals zero is necessary for leveraging other mathematical properties, particularly when preparing to factor the expression or apply the zero product property.

Zero Product Property

The zero product property is a fundamental concept that states if the product of two factors is zero, then at least one of the factors must be zero. This property is crucial when solving quadratic equations through factoring.

Here's why it's important:

• Once a quadratic is factored into two expressions \((p)(q) = 0\), this property allows us to set each factor equal to zero.
• This leads to simpler linear equations \( p = 0 \) or \( q = 0 \) that can be easily solved for their roots.
• It essentially breaks down complex quadratic equations into manageable parts.

In practice, the zero product property enables the transformation of an unmanageable quadratic into straightforward linear factors, providing a clear pathway to finding the solutions (or roots) that satisfy the original equation. It highlights the importance of getting the quadratic equation into a factored form first, which is why the factoring method pairs seamlessly with this property.