Problem 17 - Algebra for College Students

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

The given quadratic equation is \(25x^2 - 30x + 9 = 0\). This is in the standard form \(ax^2 + bx + c\), where \(a = 25\), \(b = -30\), and \(c = 9\).

Step 2: Check for a Perfect Square Trinomial

To factor, we check if the quadratic is a perfect square trinomial. A perfect square trinomial has the form \((ax - b)^2 = a^2x^2 - 2abx + b^2\). Compare and identify values such that \(a^2 = 25\), \(2ab = 30\), and \(b^2 = 9\). Here, \(a = 5\) and \(b = 3\).

Step 3: Write the Equation as a Perfect Square

The equation \(25x^2 - 30x + 9\) can be rewritten as \((5x - 3)^2 = 0\). This is because \(25x^2\) is \((5x)^2\), \(9\) is \(3^2\), and \(-30x\) is \(-2 \times 5x \times 3\).

Step 4: Apply the Zero Product Property

The zero product property states that if \(ab = 0\), then \(a = 0\) or \(b = 0\). Here, \((5x - 3)^2 = 0\) implies \(5x - 3 = 0\).

Step 5: Solve for x

Solving \(5x - 3 = 0\), add 3 to both sides to get \(5x = 3\). Then divide by 5 to find \(x\): \(x = \frac{3}{5}\).

Key Concepts

Factoring Techniques

Factoring plays a pivotal role in solving quadratic equations. It involves breaking down an expression into simpler terms, known as factors, that when multiplied together give the original expression. In this context, we aim to convert a quadratic equation such as \(25x^2 - 30x + 9 = 0\) into a format that can be easily solved.

• First, we check if the quadratic can be expressed as a product of simpler binomials.
• This involves checking if the quadratic is a perfect square trinomial, a special case of factoring.
• Factoring techniques rely on identifying specific patterns or using methods like the AC method or grouping to simplify quadratic expressions.

Recognizing these patterns is key to rewriting the original equation in a factorable form.

Perfect Square Trinomials

A perfect square trinomial is a specific kind of polynomial that factors neatly into the square of a binomial. It appears in the form of \((ax - b)^2\) and when expanded, it matches the structure \(a^2x^2 - 2abx + b^2\).

• Firstly, identify if the first and last terms of the quadratic, \(ax^2\) and \(c\), are perfect squares themselves.
• Then, check if the middle term, \(bx\), matches the specific form \(-2abx\).
• In the given quadratic \(25x^2 - 30x + 9 = 0\), we find that \((5x)^2 = 25x^2\) and \(3^2 = 9\), and \(-2 \times 5x \times 3 = -30x\).

This confirms that the quadratic is a perfect square trinomial, allowing us to write it as \((5x - 3)^2\). This simplification is crucial for applying the zero product property effectively.

Zero Product Property

The zero product property is a fundamental concept in solving equations involving factored polynomials. It asserts that if a product of multiple terms equals zero, then at least one of the terms must equal zero.

• If we have a product like \((a)(b) = 0\), it implies either \(a = 0\) or \(b = 0\).
• This property is especially useful for solving quadratic equations factored into two binomials.
• For example, after factoring \(25x^2 - 30x + 9\) as \((5x - 3)^2 = 0\), we apply the zero product property by setting \(5x - 3 = 0\).
• Solving this equation leads us to determine that \(x = \frac{3}{5}\).

This property streamlines the path to the solution by breaking down complex expressions into manageable equations.