Problem 18 - 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. $$ 16 x^{2}-8 x+1=0 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{4}\).

Step 1: Recognize the Quadratic Form

The given equation is in the standard quadratic form as \( ax^2 + bx + c = 0 \). Here, \( a = 16 \), \( b = -8 \), and \( c = 1 \). The goal is to factor this quadratic equation.

Step 2: Factor the Quadratic Equation

The equation \( 16x^2 - 8x + 1 = 0 \) is a perfect square trinomial. This means it can be written as the square of a binomial. Here, the equation is \((4x - 1)^2 = 0\).

Step 3: Apply the Zero Product Property

The zero product property states that if \( ab = 0 \), then \( a = 0 \) or \( b = 0 \). Apply this to \((4x - 1)^2 = 0\), giving \(4x - 1 = 0\).

Step 4: Solve for \(x\)

Solve the equation \(4x - 1 = 0\) by isolating \(x\). Add 1 to both sides: \(4x = 1\). Divide both sides by 4 to get \(x = \frac{1}{4}\).

Key Concepts

Understanding Perfect Square Trinomials

A perfect square trinomial is special in the world of quadratic expressions. It's formed when a binomial, which is simply something like \((ax + b)\), is multiplied by itself. When expanded, it takes the form \(a^2x^2 + 2abx + b^2\). For instance, in your equation, \(16x^2 - 8x + 1 = 0\), notice how it resembles the form of \( (4x - 1)^2 \). This means the trinomial can be expressed as a binomial squared.

Perfect square trinomials are neat because they allow us to simplify the factoring process. Recognizing them can save you time and effort because once identified, you can skip to expressing it as a square of a binomial.

• The first term, \(16x^2\), is \((4x)^2\).
• The last term, \(1\), is \(1^2\).
• The middle term, \(-8x\), matches \(2 \times 4x \times 1\).

This symmetry confirms the trinomial is indeed a perfect square, which can be factored as \((4x - 1)^2 = 0\). Recognizing this property speeds up the journey to solving the equation.

Applying the Zero Product Property

The zero product property is a cornerstone in solving equations involving products. It's a straightforward idea: if you have a product of numbers, say \(a \cdot b\), that equals zero, then either \(a = 0\) or \(b = 0\) (or both). This property is incredibly useful for quadratic equations that have been factored.

In this problem, after recognizing that \(16x^2 - 8x + 1 = (4x - 1)^2\), we see that the equation becomes \((4x - 1)^2 = 0\). To apply the zero product property:

• Understand that since \((4x - 1)^2 = (4x - 1) \cdot (4x - 1)\), each factor must be zero for their product to be zero.
• Therefore, set \(4x - 1 = 0\).

This method turns the problem into a simple linear equation, paving the way for the final solution.

Solving for \(x\)

The final step involves solving a basic linear equation. Once you have used the zero product property and set \(4x - 1 = 0\), the equation simplifies to something quite manageable. Solving it will give you the value of \(x\).

• Start by isolating \(4x\): add 1 to both sides to get \(4x = 1\).
• Next, divide both sides by 4 to isolate \(x\) yielding \(x = \frac{1}{4}\).

This result is the solution to the original quadratic equation. Every step in this process—from identifying the perfect square trinomial to employing the zero product property—points towards this solution, demonstrating a logical and structured way to solve quadratics through factoring.