Problem 15 - Elementary Algebra

Question

Solve. $$ (x-12)(x+34)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 12\) and \(x = -34\).

Step 1: Identify the Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, then at least one of the factors must be zero. We apply this property to the equation \((x-12)(x+34)=0\).

Step 2: Set Each Factor to Zero

According to the Zero Product Property, either \(x-12 = 0\) or \(x+34 = 0\). We will solve each of these equations separately to find the solutions for \(x\).

Step 3: Solve for x in Each Equation - Equation 1

First, solve \(x-12=0\). Add 12 to both sides to isolate \(x\): \(x = 12\).

Step 4: Solve for x in Each Equation - Equation 2

Now solve \(x+34=0\). Subtract 34 from both sides to isolate \(x\): \(x = -34\).

Step 5: Write the Solution Set

The solutions to the equation \((x-12)(x+34)=0\) are the values of \(x\) that satisfy each of the individual equations. Thus, the solution set is \(x=12\) and \(x=-34\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest exponent of the variable, usually denoted as \( x \), is 2. A standard form of this can be represented as \( ax^2 + bx + c = 0 \). When solving these equations, we're seeking values of \( x \) that make the equation true, known as the roots of the equation. Most problems involving quadratic equations can be solved using several methods, including factorization, completing the square, or applying the quadratic formula.

In particular, equations in the format given in the exercise involve recognizing and applying the Zero Product Property. This property comes into play specifically when you have a factorable equation, meaning you can express the quadratic in terms of its factors, such as \((x-12)(x+34)=0\). This technique simplifies the solving process because once factored, each part sets up a straightforward linear equation to solve.

Factorization

Factorization is a vital mathematical technique often used to simplify polynomials by expressing them as a product of simpler factors. In dealing with quadratic equations, factorization involves breaking down the quadratic expression into the product of two binomials. The beauty of factorization lies in its compatibility with the Zero Product Property, which states that if any product equals zero, one or both of the factors must be zero.

When faced with a factorable equation like \((x-12)(x+34)=0\), factorization has already been achieved. Here, the quadratic has turned into a product of two binomial expressions. It essentially transforms the quadratic into two smaller and more manageable equations to solve. These are linear equations and are solved much more easily compared to the quadratic format.

Solution Set

The solution set of an equation is the collection of values that satisfy the equation. For the quadratic equation expressed in factorable form as \((x-12)(x+34)=0\), we determine the solution set using the Zero Product Property.

By setting each factor equal to zero, we solve for \( x \) in each equation separately:

• From \( x-12 = 0 \), we find \( x = 12 \).
• From \( x+34 = 0 \), we find \( x = -34 \).

These solutions mean that when \( x \) is either 12 or -34, the original equation equals zero. Therefore, the solution set is \( \{ x=12, x=-34 \} \). Understanding this concept ensures that you can confidently solve any quadratic equation presented in factorized form.