Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation algebraically. Then write the equation in the form \(f(x)=0\) and use a graphing utility to verify the algebraic solution. $$(x+1)^{2}+2(x-2)=(x+1)(x-2)$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = \frac{1}{5} \). The graph of \( f(x) = 5x - 1 \) confirms this, as it crosses the x-axis at \( x = \frac{1}{5} \).

Step 1: Expand the equations

Expanding the equations gives us \( x^{2} + 2x + 1 + 2x - 4 = x^{2} - x - 2 \)

Step 2: Simplify the equation

By combining similar terms, equation simplifies to \( x^{2} + 4x - 3 = x^{2} - x - 2 \)

Step 3: Isolate terms involving x

Next subtract \( x^{2} \) and \( -x \) from both sides to get: \( 5x - 1 = 0\)

Step 4: Solve for x

Solving this equation for x, we get the solution \( x = \frac{1}{5} \)

Step 5: Verify using a graphing utility

Plot the function \( f(x) = 5x - 1 \). Use a graphing tool to verify that the function equals zero at the point \( x = \frac{1}{5} \)

Key Concepts

Expanding Equations

Expanding equations means rewriting them to show each term clearly. This usually involves using the distributive property or the FOIL method for binomials. In our problem, we have expressions like \((x+1)^2\) and \((x+1)(x-2)\).

• For \((x+1)^2\), you apply the rule \(a^2 = a \cdot a\). This gives \(x^2 + 2x + 1\).
• For \(2(x-2)\), distribute the 2 to each term inside the parentheses, resulting in \(2x - 4\).
• Finally, for \((x+1)(x-2)\), use the FOIL method: First, Outer, Inner, Last, which gives \(x^2 - x - 2\).

Breaking down each part carefully helps you see all terms and make integration into one equation easier.

Simplifying Equations

Once expanded, the next step is simplifying the equations. This means combining like terms to make the equation shorter and more manageable. For example, from our expansion of the left-hand side, we have:

• \(x^2\), \(2x\), and another \(2x\), which are like terms.
• Add them to get \(4x\).
• Then consolidate constants \(1\) and \(-4\), resulting in \(-3\).

So, from the expanded equation, you simplify to \(x^2 + 4x - 3 = x^2 - x - 2\). Removing duplicate terms, like \(x^2\), leaves us with \(5x - 1 = 0\). This simplification is crucial because it makes solving for the variable much cleaner.

Graphing Utility

A graphing utility is a tool that helps visualize mathematical functions. You can use online graphing calculators or apps to draw the graph of a function. In this case, the final equation after simplifying is \(f(x) = 5x - 1\).

• When you input \(f(x) = 5x - 1\) into the graphing tool, it shows the graph of this linear function.
• The point where the graph intersects the x-axis is the solution to the equation \(5x - 1 = 0\).
This visualization is powerful because it confirms the algebraic solution and provides a visual interpretation of the equation's root.

Solving for x

Solving for \(x\) means finding the value of \(x\) that makes the equation true. After simplifying, you have \(5x - 1 = 0\).

• To solve for \(x\), begin by adding 1 to both sides: \(5x = 1\).
• Then, divide both sides by 5 to isolate \(x\): \(x = \frac{1}{5}\).

This operation isolates the variable \(x\) on one side of the equation. Solving equations this way is often the final step after expanding and simplifying. It's like piecing together clues to find that single value that satisfies the entire equation.