Problem 28 - Algebra 2 and Trigonometry

Question

Solve for the variable and check. Each solution is an integer. \(4 x(x+2)-x(3+4 x)=2 x+18\)

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Expand the equation

First, we need to distribute the terms in the equation. Start with the left side:\[ 4x(x+2) - x(3+4x) = 2x + 18 \]Expand the terms:\[ 4x^2 + 8x - 3x - 4x^2 = 2x + 18 \]

Step 2: Combine like terms

Now, combine the like terms on the left side of the equation:\[ 4x^2 - 4x^2 + 8x - 3x = 2x + 18 \]This simplifies to:\[ 5x = 2x + 18 \]

Step 3: Isolate the variable

To solve for \(x\), get all \(x\) terms on one side by subtracting \(2x\) from both sides:\[ 5x - 2x = 18 \]\[ 3x = 18 \]

Step 4: Solve for x

Divide by 3 to solve for \(x\):\[ x = \frac{18}{3} \]\[ x = 6 \]

Step 5: Check the solution

Substitute \(x = 6\) back into the original equation to verify.Original equation:\[ 4x(x+2) - x(3+4x) = 2x + 18 \]Substitute \(x=6\):\[ 4(6)(6+2) - 6(3+4(6)) = 2(6) + 18 \]Calculate each part:\[ 4(6)(8) - 6(3 + 24) = 12 + 18 \]\[ 192 - 6(27) = 30 \]\[ 192 - 162 = 30 \]\[ 30 = 30 \]The values are equal, confirming \(x=6\) is a correct solution.

Key Concepts

Expanding Expressions

Expanding expressions is an essential step in solving algebraic equations and involves removing parentheses by distributing factors. In the given problem, there's an expression that needs expanding:

• \(4x(x+2)\) becomes \(4x^2 + 8x\) after distributing \(4x\) across both terms inside the parentheses.
• On the other hand, \( -x(3+4x)\) transforms into \(-3x - 4x^2\).

This process can change the appearance of an equation, making it easier to work with. It's vital to ensure all distributed multiplication is performed accurately. This first step sets the foundation for further simplification of the equation.

Combining Like Terms

Once you've expanded the expression, the next logical step is to combine like terms. These are terms within the equation that have the same variables raised to the same power. This means:

• In our example, \(4x^2\) offsets \(-4x^2\), effectively canceling each other out.
• Next, address the linear terms: \(8x\) and \(-3x\). When combined, they simplify to \(5x\).

This step significantly simplifies the appearance of the equation, making it less cluttered and easier to manage. Combining like terms is critical as it consolidates similar components, paving the way for isolating variables next.

Isolating Variables

The ultimate goal of solving an equation is to isolate the variable, allowing you to explicitly define it. After combining like terms in the previous step, you arrive at:

• \(5x = 2x + 18\)

Here's how you isolate the variable:

• Subtract \(2x\) from both sides to simplify the equation further. This gives you \(3x = 18\).
• Next, divide by the coefficient of \(x\), which is \(3\), to fully isolate \(x\): \(x = 6\).

At this stage, you've identified the solution, which indicates the value of the variable that satisfies the original equation. Isolating the variable is crucial to reveal this solution.

Checking Solutions

Checking your solution is always a good practice to confirm accuracy. When you believe you have the correct answer, insert this value back into the original equation to verify:

• Insert \(x = 6\) into the original equation and simplify both sides.
• If both sides of the equation are equal after substitution and simplification, your solution is confirmed. In this case, \(30 = 30\).

This verification method ensures there are no calculation errors overlooked in previous steps. By substituting back into the original equation and confirming both sides balance, you validate the solution is correct, which in this case, it is: \(x = 6\) truly satisfies the equation.