Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-14: Solve the equation by factoring. $$ 12 x^{2}+60 x+75=0 $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{5}{2}\).

Step 1: Identify and extract the greatest common factor (GCF)

First, identify the greatest common factor of all terms in the equation. Here, the GCF of 12, 60, and 75 is 3. Factor out this GCF from each term of the quadratic equation:\[3(4x^2 + 20x + 25) = 0\]

Step 2: Simplify by dividing out the GCF

Once we have factored out the GCF, divide both sides of the equation by 3 to simplify:\[4x^2 + 20x + 25 = 0\]

Step 3: Recognize and set up for perfect square trinomial factoring

Notice that the trinomial can potentially be a perfect square. The expression can be written as:\[(2x + 5)^2 = 0\]

Step 4: Set the factor equal to zero and solve for x

Since \((2x + 5)^2 = 0\), the solution to the equation is when:\[2x + 5 = 0\]Subtract 5 from both sides:\[2x = -5\]Divide by 2:\[x = -\frac{5}{2}\]

Key Concepts

Greatest Common Factor

The concept of the greatest common factor (GCF) is fundamental to simplifying quadratic equations. The GCF is the largest number that divides evenly into each coefficient of the terms in a polynomial. For the equation \(12x^2 + 60x + 75 = 0\), the coefficients are 12, 60, and 75. To find the GCF, we look for the largest number that can divide these coefficients without leaving a remainder. In this case, the GCF is 3.

Here's how it helps:

• By factoring out the GCF, the equation is simplified to a more manageable form. This makes the next steps in the solving process easier.
• Extracting the GCF reduces the equation to \(3(4x^2 + 20x + 25) = 0\) and simplifies further when divided by 3 to \(4x^2 + 20x + 25 = 0\).

Extracting the GCF effectively breaks down the problem, allowing us to focus on the core quadratic equation. Breaking large numbers into smaller components helps not only in arithmetic but also in maintaining accuracy during the problem-solving process.

Perfect Square Trinomial

A perfect square trinomial is an expression of the form \((ax + b)^2\) which expands to \(a^2x^2 + 2abx + b^2\). Recognizing a perfect square trinomial can greatly simplify the factoring process. For example, the equation \(4x^2 + 20x + 25 = 0\) can be rearranged into a perfect square trinomial.

Why is this useful?

• Perfect square trinomials allow us to convert a quadratic expression into a squared binomial, in this case turning \(4x^2 + 20x + 25\) into \((2x + 5)^2\).
• It reflects a direct method of factoring that leads straight to finding the roots of the equation.

By recognizing the pattern in the trinomial, we used the formula \[a^2x^2 + 2abx + b^2 = (ax + b)^2\], where \(a = 2\) and \(b = 5\). This leads us to \((2x + 5)^2 = 0\), paving the way for a streamlined solution.

Solving Quadratic Equations

Solving quadratic equations often involves finding the values of \(x\) that satisfy the equation. Once an equation is factored, especially using methods like finding a perfect square, the next step is using the zero-product property. For \((2x + 5)^2 = 0\), we know that for the product to be zero, the expression inside the square must also be zero.

Steps to solve:

• First, set the inside of the squared form equal to zero: \(2x + 5 = 0\).
• Isolate \(x\) by subtracting 5 from both sides to get \(2x = -5\).
• Finally, divide by 2 to solve for \(x\), resulting in \(x = -\frac{5}{2}\).

This method allows us to directly find the solution to the equation. Quadratic equations often have two solutions, but perfect squares lead to a single repeated solution, simplifying our solution process.