Problem 84 - Algebra A Combined Function

Question

Solve each equation. $$ (2 x-3)(x+6)=(x-9)(x+2) $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -16\).

Step 1: Expand the Left Side

First, we expand the left-hand side of the equation \((2x - 3)(x + 6)\). We distribute each term of \(2x - 3\) into each term of \(x + 6\):\[ (2x - 3)(x + 6) = 2x \cdot x + 2x \cdot 6 - 3 \cdot x - 3 \cdot 6 \]Simplifying, we get:\[ 2x^2 + 12x - 3x - 18 = 2x^2 + 9x - 18 \]

Step 2: Expand the Right Side

Now, expand the right-hand side \((x - 9)(x + 2)\). Distribute each term:\[ (x - 9)(x + 2) = x \cdot x + x \cdot 2 - 9 \cdot x - 9 \cdot 2 \]Simplifying, we find:\[ x^2 + 2x - 9x - 18 = x^2 - 7x - 18 \]

Step 3: Set Equations Equal

The expanded forms of both sides of the equation are now:\[ 2x^2 + 9x - 18 = x^2 - 7x - 18 \]

Step 4: Move All Terms to One Side

To solve for \(x\), we move all terms to one side of the equation to set it to zero. We subtract \(x^2 - 7x - 18\) from both sides:\[ 2x^2 + 9x - 18 - x^2 + 7x + 18 = 0 \]Simplifying, this gives:\[ x^2 + 16x = 0 \]

Step 5: Factor the Equation

Factor out the common term \(x\):\[ x(x + 16) = 0 \]

Step 6: Solve for x

Set each factor equal to zero:1. \(x = 0\)2. \(x + 16 = 0\) which gives \(x = -16\).

Step 7: Conclusion

The solutions to the equation \((2x - 3)(x + 6) = (x - 9)(x + 2)\) are \(x = 0\) and \(x = -16\).

Key Concepts

Polynomial Expansion

Polynomial expansion involves multiplying two expressions to remove parentheses and simplify the equation. It plays a critical role in understanding quadratic equations. By expanding polynomials, you can see all terms individually, making it easier to simplify and solve the equation.

In our equation \[(2x - 3)(x + 6) = (x - 9)(x + 2),\] we begin by expanding each side separately. For the left side, we multiply each term of \(2x - 3\) by each term of \(x + 6\), using the distributive property:

• First, multiply \(2x\) by \(x\).
• Next, multiply \(2x\) by \(6\).
• Then, multiply \(-3\) by \(x\).
• Finally, multiply \(-3\) by \(6\).

After performing these multiplications, we combine like terms to get the expanded expression \(2x^2 + 9x - 18\).

Repeat similar steps for the right side. Expanding \((x - 9)(x + 2)\) results in the expression \(x^2 - 7x - 18\).

This careful expansion reveals a simpler form of the polynomial, laying the foundation for solving the equation.

Factoring

Factoring is the process of writing a polynomial as a product of its factors. It's essential for solving quadratic equations because it allows us to find the equation's roots, or solutions, by setting each factor equal to zero.

In our case, after expanding both sides of the equation and simplifying, we obtain the expression: \[x^2 + 16x = 0.\] The key to factoring is to look for common terms.

• Notice that both terms share a common factor of \(x\).
• Factoring out \(x\) gives us \(x(x + 16) = 0\).

If a term, like \(x\) in this expression, is factored out, you're left with a simpler expression to work with.

Factoring also plays a crucial role in finding the solutions of the equation, as each solution corresponds to a factor being zero. This method greatly simplifies the process of identifying all possible solutions.

Solving Equations

Solving equations, particularly quadratic ones, involves several steps to isolate the variable and find its value. The solved example \((2x - 3)(x + 6) = (x - 9)(x + 2)\) illustrates how we approach this.

Begin by expanding and simplifying both sides until they're set equal. Then, collect all terms on one side to achieve the goal of forming a quadratic equation set to zero:\[2x^2 + 9x - 18 = x^2 - 7x - 18\] becomes \[x^2 + 16x = 0.\]

The next step is factoring, revealing the products of terms that equate to zero. When you have something in the form \(x(x + 16) = 0\), solving involves setting each factor equal to zero:

• \(x = 0\)
• \(x + 16 = 0\), which simplifies to \(x = -16\)

This approach not only reduces complexity but also systematically uncovers all possible values for \(x\), providing a comprehensive solution to the equation.

Distributive Property

The distributive property is an essential mathematical principle allowing us to simplify expressions by multiplying a single term outside the parentheses by each term inside.

In equations like \((2x - 3)(x + 6)\), the distributive property helps dismantle polynomial expressions into their parts. Here's how it works step-by-step:

• Multiply the first term outside the parentheses by each term within, e.g., \(2x \times x\) and \(2x \times 6\).
• Similarly, multiply the second term outside by each inner term, e.g., \(-3 \times x\) and \(-3 \times 6\).

Each multiplication results in new expressions that are the building blocks for the fully expanded polynomial.

The beauty of the distributive property lies in its simplicity and power to break down complex operations into straightforward calculations, making it easier to handle and solve equations.