Problem 32 - Algebra A Combined Function

Question

Solve.$$ (x+3)(x+8)=x $$

Step-by-Step Solution

Answer

The solutions are \( x = -4 \) and \( x = -6 \).

Step 1: Expand the Left Side of the Equation

We start by expanding the left-hand side of the equation using the distributive property. The equation is: \( (x+3)(x+8) = x \).Use the distributive property: \[(x+3)(x+8) = x(x+8) + 3(x+8) = x^2 + 8x + 3x + 24 \].Combine like terms: \[x^2 + 11x + 24 \].This is the expanded form of the left side.

Step 2: Rearrange the Equation

Subtract \( x \) from both sides of the equation to bring all terms to one side:\[ x^2 + 11x + 24 = x \]Subtract \( x \) from both sides:\[ x^2 + 11x + 24 - x = 0 \]This simplifies to:\[ x^2 + 10x + 24 = 0 \].Now we have a quadratic equation to solve.

Step 3: Solve the Quadratic Equation

To solve the quadratic equation \( x^2 + 10x + 24 = 0 \), we can factor it. We need two numbers that multiply to 24 and add to 10.The numbers 4 and 6 work because \( 4 \times 6 = 24 \) and \( 4 + 6 = 10 \).So we can factor the quadratic as:\( (x + 4)(x + 6) = 0 \).

Step 4: Find the Solutions

Set each factor equal to zero to solve for \( x \):1. \( x + 4 = 0 \) gives \( x = -4 \).2. \( x + 6 = 0 \) gives \( x = -6 \).These are the two solutions to the equation.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept that allows us to simplify expressions and solve equations effectively. It's like a handy tool in a tailor's shop for cutting fabric into perfect pieces. When we encounter a scenario such as \((x+3)(x+8)\), the distributive property helps us break it down.To use the distributive property:

• Distribute each term in the first binomial across all the terms in the second binomial.
• First, distribute \(x\) across \(x+8\), which gives us \(x \cdot x + x \cdot 8 = x^2 + 8x\).
• Next, distribute \(3\) across \(x+8\), leading to \(3 \cdot x + 3 \cdot 8 = 3x + 24\).

All these parts collectively give us the expanded form \(x^2 + 8x + 3x + 24\). Using this property simplifies complex expressions into something more manageable.

Factoring Quadratics

Factoring quadratics can be thought of as reversing the expansion process. We're putting a jigsaw puzzle back together with the pieces provided by the quadratic equation. After simplifying the equation to \(x^2 + 10x + 24 = 0\), the next step is factoring.For factoring:

• We identify two numbers that multiply to the constant term, in this case, 24, and add to the coefficient of \(x\), which is 10.
• The numbers we find are 4 and 6 because \(4 \times 6 = 24\) and \(4 + 6 = 10\).
• This means the quadratic can be expressed as \((x + 4)(x + 6) = 0\).

Factoring simplifies solving the quadratic as we transform it into a simple form, where each factor can be solved separately for \(x\).

Combining Like Terms

Combining like terms is a method used to simplify expressions by merging terms with the same variables and exponents. It's like tidying up your desk by putting all pencils in one cup and all papers in one stack. By organizing, we make the equation easier to manage and solve.Let's take the expanded form \(x^2 + 8x + 3x + 24\). To combine like terms:

• Notice the terms \(8x\) and \(3x\) are similar, as they both contain the variable \(x\).
• Add them together: \(8x + 3x = 11x\).
• Therefore, the expression simplifies to \(x^2 + 11x + 24\).

This step paves the way for rearranging the equation or further simplification.

Rearranging Equations

Rearranging equations is like moving furniture in a room to create more space or achieve better functionality. It involves manipulating the equation to get all terms on one side and zero on the other, setting the stage for identifying solutions.Here's how we rearrange in our context:

• We started with \(x^2 + 11x + 24 = x\).
• To have a standard form of a quadratic equation, shift \(x\) from the right side to the left by subtracting \(x\) from both sides.
• This yields \(x^2 + 11x + 24 - x = 0\), simplifying to \(x^2 + 10x + 24 = 0\).

Rearranging creates a clearer path toward using factoring or other methods to solve the equation.