Problem 23 - Basic Technical Mathematics with Calculus

Question

Use a calculator to solve the given equations. If there are no real roots, state this as the answer. \(6 R^{2}=18-7 R\)

Step-by-Step Solution

Answer

The quadratic equation has two real roots: approximately 1.24 and -2.41.

Step 1: Rearrange the Equation

First, rewrite the equation to bring all terms to one side. Start with the original equation: \[ 6R^2 = 18 - 7R \]Subtract \(18\) and add \(7R\) to both sides to get:\[ 6R^2 + 7R - 18 = 0 \]

Step 2: Identify Quadratic Coefficients

Identify the coefficients in the quadratic equation \(6R^2 + 7R - 18 = 0\). These are:- \(a = 6\)- \(b = 7\)- \(c = -18\)

Step 3: Use the Quadratic Formula

Use the quadratic formula \( R = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \(R\). Plug in the values of \(a\), \(b\), and \(c\):\[ R = \frac{-7 \pm \sqrt{7^2 - 4 \times 6 \times (-18)}}{2 \times 6} \]

Step 4: Calculate the Discriminant

Calculate the discriminant \(D = b^2 - 4ac\):\[ D = 7^2 - 4 \times 6 \times (-18) \]\[ D = 49 + 432 \]\[ D = 481 \]Since 481 is positive, there are two real roots.

Step 5: Compute the Roots

Using the quadratic formula, compute the roots:\[ R = \frac{-7 \pm \sqrt{481}}{12} \]Calculate each possibility:\[ R_1 = \frac{-7 + \sqrt{481}}{12}, \quad R_2 = \frac{-7 - \sqrt{481}}{12} \]

Step 6: Use a Calculator to Find the Values

Use a calculator to find the decimal values of the roots. Compute the values:\[ R_1 \approx \frac{-7 + 21.93}{12} \approx 1.24\]\[ R_2 \approx \frac{-7 - 21.93}{12} \approx -2.41\]

Key Concepts

Quadratic Formula

The quadratic formula is a cornerstone in solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides a reliable way to find the roots, or solutions, when the equation can't be easily factored. The formula is given by: \[R = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] where

• \(a\), \(b\), and \(c\) are the coefficients of the equation,
• \(b^2 - 4ac\) is part of the formula known as the discriminant,
• \(\pm\) indicates there could be two solutions: one adding and the other subtracting the square root.

Using this formula allows you to determine the roots systematically, even if they are complex or not easy to factor out. It is important to first rearrange your equation so that all terms are on one side before applying the formula.

Discriminant

The discriminant is a crucial component of the quadratic formula. It is found under the square root in the formula:\[D = b^2 - 4ac\] Understanding the value of the discriminant helps in predicting the nature of the roots:

• If \(D > 0\), there are two distinct real roots, meaning the graph of the quadratic equation will intersect the x-axis at two points.
• If \(D = 0\), there is exactly one real root, or a repeated root, and the graph touches the x-axis at one point.
• If \(D < 0\), there are no real roots, as the graph does not cross or touch the x-axis. This indicates the roots are complex numbers.

In our example, the discriminant is calculated as 481, which is greater than zero, indicating two real roots.

Real Roots

Real roots refer to the x-values where the graph of a quadratic equation crosses the x-axis. A quadratic equation can have:

• No real roots if the discriminant is less than zero; in this case, the solutions are complex.
• One real root if the discriminant is zero; the parabola touches the x-axis at exactly one point.
• Two real roots if the discriminant is greater than zero; the parabola intersects the x-axis at two points.

In practical terms, solving for real roots using the quadratic formula involves calculating the specific values of \(R\) by plugging in the determined coefficients. In our problem, the roots are \(R_1 \approx 1.24\) and \(R_2 \approx -2.41\), both of which are real numbers.

Calculator Usage

Using a calculator for solving the quadratic equation is immensely helpful, especially when dealing with non-perfect square discriminants or large numbers. Here are some tips when using a calculator in this scenario:

• Ensure your calculator is in the correct mode for operations—in most cases, standard or real number mode.
• Step by step, input values from the quadratic formula, especially when dealing with complex under-the-root calculations.
• First compute the discriminant, log the result, and then proceed with finding the roots.
• Divide the computation into manageable parts to avoid errors, calculate \(b^2\), \(-4ac\), and the whole \(\sqrt{481}\) separately before putting them together.
• Use the memory functions of the calculator to store intermediate results if available.

By following a systematic approach, you ensure accurate results—just as with our calculated roots of \(R_1\approx 1.24\) and \(R_2\approx -2.41\). Always double-check your computations, as even small mistakes can lead to incorrect answers.