Problem 18 - Algebra A Combined Function

Question

Solve each equation. $$ (x+1.7)(x+2.3)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -1.7\) and \(x = -2.3\).

Step 1: Identify the Zero Product Property

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. This means either \((x + 1.7) = 0\) or \((x + 2.3) = 0\).

Step 2: Solve the First Equation

Set the first factor equal to zero: \(x + 1.7 = 0\). Then solve for \(x\) by subtracting 1.7 from both sides. Thus, \(x = -1.7\).

Step 3: Solve the Second Equation

Set the second factor equal to zero: \(x + 2.3 = 0\). Then solve for \(x\) by subtracting 2.3 from both sides. Thus, \(x = -2.3\).

Step 4: Verify Solutions

Plug the solutions back into the original equation to ensure they satisfy the equation. For \(x = -1.7\), the equation becomes \((-1.7 + 1.7)(-1.7 + 2.3) = 0\), which simplifies to \(0 \times 0.6 = 0\). Similarly, for \(x = -2.3\), \((-2.3 + 1.7)(-2.3 + 2.3) = 0\), simplifies to \(-0.6 \times 0 = 0\). Both solutions satisfy the original equation.

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental concept used in algebra, especially when dealing with quadratic equations. At its core, it simply means that if the product of two numbers (or expressions) is zero, then at least one of those numbers must be zero. This is logical because multiplying any number by zero results in zero.

For the equation \[(x+1.7)(x+2.3)=0\], the Zero Product Property allows us to break it into two separate equations to solve. Each factor of the equation \((x+1.7)\) and \((x+2.3)\) can independently be zero. Thus, we set each factor to zero to find possible solutions for \(x\).

• The first case: \(x + 1.7 = 0\)
• The second case: \(x + 2.3 = 0\)

By solving these simple equations, we can find the values of \(x\) that satisfy the original quadratic equation. This property is powerful because it simplifies the process of finding solutions, making complex equations more manageable.

Solving Equations

Solving equations is one of the primary tasks in algebra. It involves finding the values of variables that make the equation true. In our problem, we dealt with a quadratic equation derived from an expression set equal to zero, \((x+1.7)(x+2.3)=0\).

The goal is to isolate the variable \(x\) on one side of the equation. Using the Zero Product Property, we split the initial equation into two simpler linear equations: \(x + 1.7 = 0\) and \(x + 2.3 = 0\). Solving each one separately involves straightforward algebraic manipulation.

• For \(x + 1.7 = 0\), we subtract \(1.7\) from both sides, yielding \(x = -1.7\).
• For \(x + 2.3 = 0\), we subtract \(2.3\) from both sides, resulting in \(x = -2.3\).

By isolating \(x\) and performing simple arithmetic, we solve each equation step by step. This systematic approach makes solving quadratic equations accessible and intuitive.

Factoring

Factoring is the process of breaking down complex expressions into products of simpler factors. This mathematical technique is especially useful in solving quadratic equations because it allows application of the Zero Product Property.

In the given exercise, the original factored form is already present:\((x+1.7)(x+2.3)\) is the factored version of the quadratic expression. This makes it easy to apply the Zero Product Property directly and find solutions quickly.

Understanding how to factor newer expressions is crucial, as not all equations come pre-factored. For complex quadratic equations, steps include:

• Identifying common factors.
• Rewriting the expression as a product of binomials or other simpler expressions.
• Setting each factor to zero to solve for unknown variables.

Once you get the hang of factoring, it becomes a powerful tool for making algebraic problems much easier to solve. Whether pre-factored or always requiring manual factoring, knowing this technique is invaluable in algebra.