Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to solve the equation. (Round your solution to three decimal places.) \((x+5.62)^{2}+10.83=(x+7)^{2}\)

Step-by-Step Solution

Answer

The solution for x is approximately -6.314

Step 1: Expand the Squares

Expand the squares on both sides of the equation. This gives: \(x^{2} + 2 \cdot x \cdot 5.62 + 5.62^{2} + 10.83 = x^{2} + 2 \cdot x \cdot 7 + 7^{2}\). Now simplify to: \(x^{2} + 11.24x + 31.58 = x^{2} + 14x + 49\).

Step 2: Simplify The Equation

Subtract \(x^{2}\) and \(11.24x\) from both sides to get the equation in form of \(ax = b\). The simplified equation becomes \(2.76x + 17.42 = 0\)

Step 3: Solve for x

Isolate x by subtracting 17.42 from both sides and then divide the result by 2.76. The final equation is \(x = -17.42 / 2.76\).

Step 4: Find x Using a Calculator

Input the equation into the calculator to find the value of x. Make sure to round your answer to three decimal places.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a critical skill in solving equations like the one presented. It involves expanding, simplifying, and rearranging equations to make them easier to solve.
In this exercise, we begin with \((x+5.62)^{2}+10.83=(x+7)^{2}\). To manipulate this, we expand both sides of the equation. This means distributing the terms within each square:

• Expanding \((x + 5.62)^2\) gives \(x^2 + 2\cdot x \cdot 5.62 + 5.62^2\).
• Similarly, \((x + 7)^2\) becomes \(x^2 + 2\cdot x \cdot 7 + 7^2\).

Now, combine these expansions in the equation, leading to \(x^2 + 11.24x + 31.58 = x^2 + 14x + 49\). Next, we simplify by eliminating the same terms on both sides, making it \(2.76x + 17.42 = 0\). With this manipulation, we set the stage for isolating and solving for \(x\).

Equation Solving

Equation solving is the process of finding unknown values that satisfy a given equation. In this context, we already have a simplified form of the equation: \(2.76x + 17.42 = 0\). The goal is to solve for \(x\), the variable in question.
We start by isolating \(x\). This means eliminating other parts of the equation. Follow these steps:

• Subtract \(17.42\) from both sides of the equation, leading to \(2.76x = -17.42\).
• Then, divide all sides by \(2.76\) to isolate \(x\), resulting in \(x = \frac{-17.42}{2.76}\).

This systematic approach ensures all steps are logical and correctly simplify the path to a solution. With \(x\) isolated, the next phase is calculation for precision, ideally rounded to three decimal places as required.

Use of Calculator

Calculators are vital tools in solving complex algebraic equations, especially when dealing with decimals or requiring precise results. Once we have isolated \(x\) as in \(x = \frac{-17.42}{2.76}\), the next step is to compute the exact value.
Using a calculator ensures accuracy, particularly for rounded answers. Here are some steps:

• Enter \(-17.42\) into the calculator, then divide by \(2.76\).
• Check entries to ensure there are no errors, as even small mistakes can lead to wrong results.
• Press the equals button to obtain the final value.

After computation, make sure to round the answer to three decimal places. This careful use of a calculator aids in verifying manual calculations and provides an exact value necessary for precise solutions.