91Ó°ÊÓ

Quadratic equations are polynomial equations of the second degree. They have the general form \[ ax^2 + bx + c = 0 \]where:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

In our problem, the quadratic expression is \[4w^2 - 5w + 1 \]Understanding the structure of quadratic equations is crucial because it helps to apply different methods to solve or factor them. Each term plays a role in the behavior and shape of the parabola represented by the quadratic equation. The coefficient \( a \) determines the width and direction of the parabola, \( b \) changes the position and vertex, while \( c \) moves the graph up or down.

Factoring by trial and error is a method used to factor quadratic expressions when they are not easily factorable by other methods. This approach requires testing different combinations of factors to simplify the expression. Let’s go through the basic steps: 1. Look at the quadratic form: \[ ax^2 + bx + c \]2. Set up the potential factored form as: \[ (mw + n)(pw + q) \]3. Determine possible pairs for \( m \) and \( p \) such that \( mp = a \) and pairs for \( n \) and \( q \) such that \( nq = c \).For our expression \[ 4w^2 - 5w + 1 \]we set \[ (4w - 1)(w - 1) \]4. Verify by expanding the pairs: \[ (4w - 1)(w - 1) = 4w^2 - 4w - w + 1 = 4w^2 - 5w + 1 \]This confirms that our factors are correct.

Once you have factored a quadratic expression, solving the equation becomes simpler. Suppose the equation is in the form \[ ax^2 + bx + c = 0 \]First, factor the quadratic expression if needed. For example, from our expression: \[ 4w^2 - 5w + 1 = 0 \] we already factored it as \[ (4w - 1)(w - 1) = 0 \] Next, use the Zero Product Property which states if \[ ab = 0 \]then either \( a = 0 \) or \( b = 0 \), you set each factor equal to zero: \[ 4w - 1 = 0 \] Solving for \( w \): \[ w = \frac{1}{4} \] and \[ w - 1 = 0 \]Solving for \( w \): \[ w = 1 \] So, the solutions to the polynomial equation \[ 4w^2 - 5w + 1 = 0 \] are \[ w = \frac{1}{4} \] and \[ w = 1 \]. Understanding this process helps in solving more complex polynomial equations.