Problem 15 - Algebra A Combined Function

Question

Solve each equation. $$ \left(x-\frac{1}{2}\right)\left(x+\frac{1}{3}\right)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = -\frac{1}{3}\).

Step 1: Understand the Zero Product Property

The equation \((x - \frac{1}{2})(x + \frac{1}{3}) = 0\) uses the zero product property, which states if a product of terms is zero, at least one of the terms must be zero. Therefore, we set each factor in the equation equal to zero.

Step 2: Solve the First Factor

Set the first factor equal to zero: \(x - \frac{1}{2} = 0\). To solve for \(x\), add \(\frac{1}{2}\) to both sides. This gives: \[x = \frac{1}{2}\].

Step 3: Solve the Second Factor

Set the second factor equal to zero: \(x + \frac{1}{3} = 0\). To solve for \(x\), subtract \(\frac{1}{3}\) from both sides. This results in: \[x = -\frac{1}{3}\].

Step 4: Verify the Solutions

Substitute each solution back into the original equation to verify. For \(x = \frac{1}{2}\): \((x - \frac{1}{2}) = 0\), the left side is zero which fits. For \(x = -\frac{1}{3}\): \((x + \frac{1}{3}) = 0\), it also validates the equation.

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental concept when working with quadratic equations. It asserts that if the product of two expressions is zero, then at least one of the expressions must itself be zero. This rule applies universally in mathematics and is particularly useful when we need to solve quadratic equations.Understanding this property allows us to deconstruct the given equation \[(x - \frac{1}{2})(x + \frac{1}{3}) = 0\]By recognizing that either - \(x - \frac{1}{2} = 0\), or- \(x + \frac{1}{3} = 0\), we can find our potential solutions. This simplifies complex equations down to manageable pieces to work with, highlighting how powerful and versatile the zero product property can be.

Factoring

Factoring is the process of breaking down an expression into a product of its simpler factors. In solving the quadratic equation \[(x - \frac{1}{2})(x + \frac{1}{3}) = 0\]factoring reveals the roots or solutions of the equation straightforwardly.When dealing with a quadratic expression, the aim is to rewrite it as a product of binomials that are simpler to solve. Here's how the expression was already factored:- The expression is directly - \((x - \frac{1}{2})\) - \((x + \frac{1}{3})\)These are the simplest forms these expressions can take to apply the zero product property effectively. In different contexts, sometimes an expression needs to be manipulated to reach this factored form.

Verifying Solutions

Verifying solutions is an essential step to ensure the solutions we've solved for are indeed correct. After finding potential solutions, \[x = \frac{1}{2}\]and \[x = -\frac{1}{3}\]we substitute them back into the original equation to check their validity. This step confirms they satisfy the equation.By plugging them back:- For \(x = \frac{1}{2}\): - Substitute into the first factor to yield - \((\frac{1}{2} - \frac{1}{2}) = 0\)- For \(x = -\frac{1}{3}\): - Substitute into the second factor to yield - \((-\frac{1}{3} + \frac{1}{3}) = 0\)Both solutions return a result of zero on their respective sides of the equation, confirming their correctness. This also showcases the importance of double-checking your work to avoid errors and solidify understanding.