Problem 15 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ 4 y^{2}-24 y+36=0 $$

Step-by-Step Solution

Answer

The solution to the quadratic equation is \(y=3\)

Step 1: Factorizing the Equation

First, factorize the quadratic equation \(4 y^{2}-24 y+36=0\). You notice that 4 is a common factor in all elements, remove it from the equation to get \((y-3)^2=0\)

Step 2: Solving for y

After factorizing the quadratic equation, you will set each factor equal to zero (since when a product is zero then at least one of the terms must be zero). Thus \((y-3)=0\)

Step 3: Finding the Solution

For \((y-3)=0\) go ahead and solve for y which gives \(y=3\)

Key Concepts

Factoring Quadratics

When we come across a quadratic equation in the form of \( ax^2 + bx + c = 0 \), our task is to transform it into a product of two binomials. The process of breaking down the equation into simpler parts—essentially finding two expressions that multiply together to give the original quadratic—is known as factoring quadratics.

To successfully factor quadratics, one typically looks for two numbers that add up to \( b \) (the coefficient of \( x \) term) and multiply to \( ac \), which is the product of \( a \) (the leading coefficient) and \( c \) (the constant). Recognizing patterns, such as perfect square trinomials and difference of squares, can greatly help in this process. In the exercise \( 4y^2-24y+36=0 \), for instance, by noticing it is a perfect square trinomial, we can factor it directly to \( (y-3)^2 \).

Here are tips to make factoring easier:

• Always look for a common factor first.
• Remember special products and patterns.
• Check your work by expanding the factors to ensure it results in the initial quadratic.

Quadratic Equations

A quadratic equation is a second-degree polynomial typically presented in the standard form \( ax^2+bx+c=0 \), where \( a \) is not equal to zero. These equations are known for their characteristic parabolic graphs and are fundamental in various areas of mathematics and science.

There are several methods for solving quadratic equations, such as

• factoring,
• using the quadratic formula,
• completing the square,
• graphing.

Choosing the best method often depends on the form of the quadratic equation and the inherent simplicity of the possible solutions. For example, in the given exercise, factoring was the most straightforward method due to the structure of the quadratic equation presented.

Zero Product Property

The zero product property is a key mathematical rule that states if the product of two factors is zero, then at least one of the factors must be zero. Formally, it can be written as, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \) (or both).

This property is especially useful when solving quadratic equations that have been factored into binomials. After factoring, as we've done in our exercise with \( (y-3)^2 = 0 \), we apply the zero product property. We set each factor equal to zero, leading us directly to the roots of the equation, which in this case is \( y = 3 \). It's a simple yet powerful tool that underscores why factoring can be a preferred method for finding the solutions to quadratic equations.