Problem 21 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by the zero-factor property. $$4 x^{2}-4 x+1=0$$

Step-by-Step Solution

Answer

x = 1/2

Step 1: Identify the Quadratic Equation

The given equation is already in the form of a standard quadratic equation: 4x^2 - 4x + 1 = 0

Step 2: Use the Zero-Factor Property

To use the zero-factor property, we need to factor the quadratic equation. Notice that this quadratic equation can be rewritten as: (2x - 1)^2 = 0

Step 3: Solve for x

Since the square of any real number is zero only if the number itself is zero, we set the factor equal to zero: 2x - 1 = 0

Step 4: Isolate x

Solve for x by isolating it on one side of the equation:2x - 1 = 0 2x = 1 x = 1/2

Key Concepts

quadratic equations

Quadratic equations are a type of polynomial equation of degree 2. They are generally written in the form:
ax^2 + bx + c = 0
In this form, 'a', 'b', and 'c' are constants, and 'x' is the variable.

What makes quadratic equations interesting is that they can have 0, 1, or 2 real roots. These roots are the values of 'x' that make the equation true.
For example, in our exercise:

4x^2 - 4x + 1 = 0
a = 4, b = -4, and c = 1.

Quadratic equations appear in various real-life situations, such as physics problems, profit calculations, and projectile motion. Understanding how to solve them is crucial for many aspects of science and engineering.

factoring

Factoring is a method used to break down complex expressions into simpler ones, making it easier to solve equations.
In our given quadratic equation:

4x^2 - 4x + 1 = 0
We can rewrite it by noticing it fits perfectly into a square of a binomial:
(2x - 1)^2 = 0

To factor a quadratic equation, look for patterns or use methods like:

• Finding two numbers that multiply to ‘ac’ and add to ‘b’
• Using special factoring formulas (e.g., the difference of squares)
• Completing the square

Factoring reveals the roots of the equation, making it easier to solve. This step is essential for applying the zero-factor property.

solving equations

Solving equations involves finding the value(s) that make the equation true. For our quadratic equation:

(2x - 1)^2 = 0

We apply the zero-factor property, which states:

If a product of factors equals zero, at least one of the factors must be zero.
Hence, we set the factor equal to zero:

2x - 1 = 0

To solve for 'x', isolate it on one side:

2x = 1
x = 1/2

By solving the equation, we've found that 'x = 1/2' is the root.
Solving quadratic equations can sometimes be tricky, but with practice, it becomes easier.