91Ó°ÊÓ

Factoring is an essential skill in solving quadratic equations. When you factor an equation, you express it as the product of two or more simpler expressions. In this problem, the equation is factored as follows:
\[ w(4w + 1) = 0 \]. This means we decompose the quadratic expression into two parts: \( w \) and \( 4w + 1 \). Each one of these factors can independently equal zero.
Factoring simplifies complex equations and allows us to find solutions more easily. Always start by examining if you can factor out common terms. In this case, \( w \) is a common factor in the expression, which leads to two separate linear equations. Learning how to factor correctly will make solving quadratics much simpler!

The Zero Product Property is a powerful tool when solving equations. It states that if the product of two factors is zero, then at least one of those factors must be zero. This property is useful in many areas of mathematics including algebra and calculus. When you encounter an equation like
\[ w(4w + 1) = 0 \], you can set each factor equal to zero:
\( w = 0 \) and \( 4w + 1 = 0 \). Solving these gives you the solutions to the equation. In our example, setting the first factor to zero, we get \( w = 0 \). Then, setting the second equation to zero and solving it:
\[ 4w + 1 = 0 \],
\[ 4w = -1 \],
\[ w = -\frac{1}{4} \]. By using the Zero Product Property, you break down a problem into simpler parts that are easier to solve.

Verification of solutions is the final and important step in solving equations. You need to ensure that the solutions obtained are correct by substituting them back into the original equation. Each possible solution should satisfy the equation. For\[ w(4w + 1) = 0 \], we found two solutions:
\( w = 0 \) and \( w = -\frac{1}{4} \). First, substitute \( w = 0 \) into the original equation:
\[ 0(4 \times 0 + 1) = 0 \],This simplifies to \( 0 = 0 \), which is true.Next, substitute \( w = -\frac{1}{4} \) into the original equation: \[ -\frac{1}{4}(4(-\frac{1}{4}) + 1) = 0 \],
\[ -\frac{1}{4}(-1 + 1) = 0 \],
\[ -\frac{1}{4} \times 0 = 0 \]. This simplifies to \( 0 = 0 \). This is also true. Verification is crucial because it confirms the accuracy of your solutions. Always make sure that each solution fits back into the original equation without contradiction.