Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation by factoring. $$ 6 x^{2}+3 x=0 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = 0\) and \(x = -0.5\).

Step 1: Rearrange and Simplify the Equation

Rearrange the equation in standard form of a quadratic equation where \( ax^2 + bx + c = 0 \), which is \(6x^2 + 3x = 0\). The simplified form by taking out the common factor \(x\) from the equation is \(x(6x + 3) = 0\).

Step 2: Solve for x

Apply the zero property rule which states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\). By applying this principle we have \(x = 0\) or \(6x + 3 = 0\). From the latter, solving for \(x\), we get \(x = -0.5\).

Step 3: State the Solution

The solutions to the quadratic equation are \(x = 0\) and \(x = -0.5\).

Key Concepts

Factoring

Factoring is one of the most essential methods in solving quadratic equations. It involves rewriting a complex equation as a product of simpler expressions. The idea here is to "factor out" the greatest common factor which in this case is "x" in the equation \[6x^2 + 3x = 0.\]This means we rewrite the equation as\[x(6x + 3) = 0.\]

The main goal is to break down the polynomial into simpler components that can be set to zero. This technique simplifies the process and directly leads us to an easier path to find the roots of the equation. Remember, when something multiplied by zero gives a zero, it implies one or more components must be zero. This ties in directly with the Zero Property Rule, which is our next step.

Zero Property Rule

The Zero Property Rule is a fundamental principle used when solving equations that have been factored. It dictates that if the product of two numbers is zero, then at least one of the numbers must be zero. In equation form, it says:

• If \(ab = 0\), then \(a = 0\) or \(b = 0\).

This rule helps us find solutions to equations quickly after factoring. Applying this rule to our earlier factored version of the equation \[x(6x + 3) = 0,\]

we see that either \(x = 0\) or \(6x + 3 = 0\). The Zero Property Rule simplifies identifying solutions by making use of the fact that anything multiplied by zero results in zero. Solving the secondary equation by isolating \(x\), we get \(x = -0.5\). This logical approach is effective in pinpointing exact and simplified solutions.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation. Factoring and the Zero Property Rule are two staple methods utilized in this process. The equation \[6x^2 + 3x = 0\]

is tackled by factoring and then applying the zero property rule. First, factor out the greatest common factor from the equation, which is "x", leading to \[x(6x + 3) = 0.\]

Next, apply the zero property rule, yielding possible solutions where \(x = 0\) or \(6x + 3 = 0\). Solve for \(x\) by isolating it, bringing us to the final solutions: \(x = 0\) and \(x = -0.5\). Understanding each step helps build a strong foundation for solving more complex quadratic equations, using this as both a method of calculation and a learning framework.