Problem 17 - Intermediate Algebra

Question

Solve each of the quadratic equations by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). If necessary, return to Chapter 3 and review the factoring techniques presented there. $$ 25 x^{2}-30 x+9=0 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{5}\).

Step 1: Identify the Quadratic Equation

We start with the quadratic equation given: \(25x^2 - 30x + 9 = 0\). Our goal is to factor this expression so that it is easier to solve.

Step 2: Factor the Quadratic Equation

Recognize that the quadratic can be factored as a perfect square trinomial. We write \( (5x - 3)^2 = 25x^2 - 30x + 9 \). Factoring gives us \((5x - 3)(5x - 3) = 0\).

Step 3: Apply the Zero Product Property

According to the zero-product property, if \( a \cdot b = 0 \), then \( a = 0 \) or \( b = 0 \). Here, since it is \((5x - 3)(5x - 3) = 0\), we set \(5x - 3 = 0\).

Step 4: Solve for x

Solve the equation \(5x - 3 = 0\) for \(x\). Add 3 to both sides to get \(5x = 3\). Then, divide each side by 5 to solve for \(x\), giving \(x = \frac{3}{5}\).

Key Concepts

Factoring Quadratics

Quadratic equations typically have the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Factoring quadratics is an essential step in solving these kinds of equations, as it simplifies them into a product of binomials. When we factor a quadratic, we're essentially rewriting it as a multiplication of two expressions.
Here's how it works in practice:

• First, identify if the quadratic is a perfect square trinomial or can be factored simply.
• If it's a perfect square trinomial like in our exercise, it will look like \( (mx + n)^2 \).
• Identifying patterns, such as perfect squares, can make factoring much quicker and easier.

The key is recognizing the structure of the quadratic expression, which can reveal how to split it into simpler components. Once it's factored, solving the equation becomes much more straightforward.

Zero Product Property

The Zero Product Property is a fundamental principle in algebra that states if a product of two factors equals zero, then at least one of the factors must be zero. Mathematically, it is expressed as:

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

This property is incredibly useful in solving quadratic equations because, after factoring, it allows us to set each factor equal to zero to find potential solutions for \( x \). For example, if we have a factored equation \((px + q)(rx + s) = 0\), we set both \( px + q = 0 \) and \( rx + s = 0 \).
Solving these simpler equations gives us the roots of the original quadratic equation. The Zero Product Property greatly simplifies the process of finding the solutions as it breaks down the equation into manageable parts.

Perfect Square Trinomial

A perfect square trinomial takes the form of \( a^2 + 2ab + b^2 \), which can be factored into \( (a + b)^2 \). Recognizing perfect square trinomials simplifies the process of factoring quadratics, making it easier to solve the equation.
In our exercise, the quadratic \( 25x^2 - 30x + 9 \) is recognized as a perfect square trinomial:

• \( 25x^2 \) is \((5x)^2\)
• \( 9 \) is \(3^2\)
• The middle term \(-30x\) matches the pattern \(2 \cdot 5x \cdot 3\), demonstrating the perfect square relationship.

Therefore, this expression can be rewritten as \((5x - 3)^2 = 0\).
By spotting this pattern, solving the quadratic becomes quicker. After factoring, you can directly apply the Zero Product Property to find the value of \( x \), simplifying the problem-solving process.