Problem 11 - Basic Engineering Mathematics

Question

Use the quadratic formula to solve \(\frac{x+2}{4}+\frac{3}{x-1}=7\) correct to 4 significant figures.

Step-by-Step Solution

Answer

The solutions are approximately 26.50 and 1.496.

Step 1: Eliminate Fractions

First, we need to eliminate the fractions. Multiply every term in the equation by the common denominator of the fractions, which in this case is \(4(x-1)\). We begin by clearing the fractions:\[4(x-1) \times \left( \frac{x+2}{4} \right) + 4(x-1) \times \left( \frac{3}{x-1} \right) = 4(x-1) \times 7\]This simplifies to:\[(x+2)(x-1) + 12 = 28(x-1)\]

Step 2: Expand and Simplify

Expand the equation by multiplying the terms:\[(x+2)(x-1) = x^2 - x + 2x - 2 = x^2 + x - 2\]Now our equation becomes:\[x^2 + x - 2 + 12 = 28x - 28\]Simplify further:\[x^2 + x + 10 = 28x - 28\]

Step 3: Rearrange into Standard Form

Move all terms to one side of the equation to set it to zero:\[x^2 + x + 10 - 28x + 28 = 0\]Combine like terms:\[x^2 - 27x + 38 = 0\]Now the equation is in standard quadratic form: \[ax^2 + bx + c = 0\], where \(a = 1\), \(b = -27\), and \(c = 38\).

Step 4: Apply the Quadratic Formula

Use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substituting for \(a\), \(b\), and \(c\):\[x = \frac{-(-27) \pm \sqrt{(-27)^2 - 4 \times 1 \times 38}}{2 \times 1}\]Simplifying inside the square root:\[b^2 - 4ac = 729 - 152 = 577\]Now substitute back:\[x = \frac{27 \pm \sqrt{577}}{2}\]

Step 5: Calculate the Final Solutions

We have two possible solutions based on \(\pm\):1. \[x = \frac{27 + \sqrt{577}}{2}\]2. \[x = \frac{27 - \sqrt{577}}{2}\]Calculate these values correct to 4 significant figures:For \(x = \frac{27 + \sqrt{577}}{2}\), compute approximately, yielding \(x \approx 26.504\).For \(x = \frac{27 - \sqrt{577}}{2}\), compute approximately, resulting in \(x \approx 1.496\).

Key Concepts

Eliminate Fractions

In algebra, equations often contain fractions, which can make them tricky to handle. To streamline the equation-solving process, it's helpful to eliminate these fractions. Here's how:

• First, determine a common denominator for all fractions in the equation.
• For the expression \( \frac{x+2}{4} + \frac{3}{x-1} = 7 \), the common denominator is \(4(x-1)\).
• Multiply every term by this common denominator. This step clears the fractions from the equation.

By applying this method, the equation transforms, making it much simpler to work with. Eliminating fractions helps in preparing the equation for further steps such as expanding and simplifying.

Expand and Simplify

Once fractions are eliminated, the next step is to expand and simplify the equation. Expanding involves multiplying out parentheses:

• The expression \((x+2)(x-1)\) is expanded to \(x^2 + x - 2\), by distributing each term.
• Once expanded, combine any like terms to simplify the equation further.
• Continue simplifying until you have a clean and manageable expression.

These actions refine the equation, cutting it down to essentials that are easier to manipulate. This process prepares the groundwork necessary for forming a standard quadratic equation.

Standard Quadratic Form

The goal of simplifying and expanding is to arrive at a standard quadratic equation. This form is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Here's how you get there:

• After expanding and simplifying, ensure all terms are on one side of the equation, setting it to zero.
• Reorganize the expression: Align similar degrees of \(x\) together.

In our example, the derived equation \(x^2 - 27x + 38 = 0\) now clearly follows this form, with \(a = 1\), \(b = -27\), and \(c = 38\). This setup is crucial for applying the quadratic formula effectively.

Calculate Significant Figures

The final piece to solving any quadratic equation is often precision. Particularly, solutions should meet specific accuracy, such as four significant figures. Here’s what this involves:

• Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Calculate both possible solutions: positive and negative square root influences.
• Determine your answer to the required number of significant figures. This means considering only important digits based on the context.

From our example, the solutions \(x \approx 26.504\) and \(x \approx 1.496\) meet the requirement of four significant figures. Accuracy at this step ensures the reliability and validity of your calculations.