Problem 55 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ (x+4)(x-8)=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -4\) and \(x = 8\).

Step 1: Apply the Zero Product Property

Because we know that the product of zero and any other number is zero, we can set each factor in the equation to zero.

Step 2: Solve for x in the First Equation

Setting the first factor equal to zero, \(x + 4 = 0\), gives the solution \(x = -4\).

Step 3: Solve for x in the Second Equation

Setting the second factor equal to zero, \(x - 8 = 0\), gives the solution \(x = 8\).

Key Concepts

Zero Product Property

The Zero Product Property is an essential tool when solving quadratic equations, especially those in factored form like \[(x+4)(x-8)=0\]. This property states that if the product of two numbers is zero, then at least one of the numbers must be zero. Thus:

• If \(a \times b = 0\), then either \(a = 0\) or \(b = 0\) (or both).

This is because zero multiplied by anything is always zero. So, when we encounter a factored equation like our example, we can set each factor to zero and solve separately.In this case, we set:

• \(x + 4 = 0\)
• \(x - 8 = 0\)

By solving these simple equations, we find the values of \(x\) that make the original product equal to zero, which are the solutions to the quadratic equation.

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions. In the context of quadratic equations, it involves rewriting a quadratic expression as the product of two binomials. For example, \[x^2 - 4x - 32\] can be factored into \[(x + 4)(x - 8)\].This makes the expression easier to solve, especially when using the Zero Product Property. By factoring a quadratic equation, you essentially set it up for easy application of this rule.In many cases, to factor an expression, you look for two numbers that multiply to the constant term (in this case, \(-32\)) and add to the coefficient of the linear term (\(-4\)). With practice, factoring becomes a straightforward process and is a critical step toward finding solutions to quadratic equations.

Solving Equations

Solving equations, especially quadratics, means finding the values of variables that satisfy the equation. Once a quadratic equation is factored, solving it by setting each factor to zero becomes straightforward.Consider the factored equation \[(x+4)(x-8)=0\].To solve it:

• Set the first factor to zero: \(x + 4 = 0\). This gives you \(x = -4\).
• Set the second factor to zero: \(x - 8 = 0\). This gives you \(x = 8\).

Thus, the equation has two solutions: \(-4\) and \(8\). Quadratic equations often have two solutions because they graph as parabolas, which can cross the x-axis at two points. By factoring and applying the Zero Product Property, we efficiently find these solutions.