Problem 29 - Algebra A Combined Function

Question

Solve. $$ x^{2}-4 x=32 $$

Step-by-Step Solution

Answer

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

Step 1: Move Everything to One Side of the Equation

To solve the equation \(x^2 - 4x = 32\), our first goal is to set it to zero. Subtract 32 from both sides to get: \(x^2 - 4x - 32 = 0\).

Step 2: Factor the Quadratic Equation

Next, factor the quadratic equation. We need to find two numbers that multiply to -32 and add to -4. These numbers are -8 and 4, so the factorization is \((x - 8)(x + 4) = 0\).

Step 3: Apply the Zero Product Property

Using the zero product property, set each factor equal to zero. For \(x - 8 = 0\), we find \(x = 8\). For \(x + 4 = 0\), we get \(x = -4\).

Step 4: Write the Solution

The solutions to the equation \(x^2 - 4x = 32\) are \(x = 8\) and \(x = -4\).

Key Concepts

Factoring

Factoring is an essential technique when solving quadratic equations. It involves breaking down a quadratic expression into simpler, multiplicative components known as factors. The goal of factoring is to express the equation in a form that reveals how the product of two simpler expressions can yield a zero value, as needed for solving.For example, consider the quadratic equation from the exercise, which we rearranged to:\[x^2 - 4x - 32 = 0\]The task is to find two numbers that multiply to -32 (the constant term) and add up to -4 (the coefficient of the middle term). These numbers are -8 and 4. Thus, the quadratic equation can be factored as:\[(x - 8)(x + 4) = 0\]Factoring transforms quadratic equations into a product of binomials, making it easier to apply further algebraic techniques. It's the precursor to using the zero product property.

Zero Product Property

The zero product property is a fundamental algebraic principle used to solve equations after they have been factored. According to this property, if the product of two numbers is zero, then at least one of the numbers must be zero.In practical terms, once we have our factored equation from before:\[(x - 8)(x + 4) = 0\]We can apply the zero product property by setting each individual factor equal to zero:- \(x - 8 = 0\)- \(x + 4 = 0\)Solving each equation separately, we find:

• \(x - 8 = 0\) leads to \(x = 8\)
• \(x + 4 = 0\) leads to \(x = -4\)

This step simplifies the solution process by breaking it down into smaller, more manageable parts.

Algebraic Solutions

Finding algebraic solutions involves combining all the methods and techniques we’ve discussed to solve the entire quadratic equation. Algebraic solutions aim to find the values of the variable that satisfy the equation from start to finish.Step by step, we moved everything to one side of the equation, making it 0, factored it, and then used the zero product property. Each piece of the solution process builds on the previous steps.In this case, once we've factored to:\[(x - 8)(x + 4) = 0\]Using the zero product property, we obtained the solutions as \(x = 8\) and \(x = -4\). These values are the algebraic solutions to our original equation. This approach not only gives us a clear path to find the solutions but also illustrates how different algebraic tools work together to solve quadratic equations efficiently.