Problem 23 - Basic Technical Mathematics with Calculus

Question

Use a calculator to solve the given equations. Round solutions to the nearest hundredth. If there are no real roots, state this. $$4 R^{2}=12-7 R$$

Step-by-Step Solution

Answer

The solutions to the equation are approximately 1.07 and -2.82.

Step 1: Reorganize the Equation

First, move all terms to one side of the equation to get a standard quadratic form. Subtract 12 and add 7R to both sides to obtain:\[ 4R^2 + 7R - 12 = 0 \]

Step 2: Identify Quadratic Coefficients

Identify the coefficients from the equation in the form \( ax^2 + bx + c = 0 \). Here, \( a = 4 \), \( b = 7 \), and \( c = -12 \).

Step 3: Calculate the Discriminant

Use the formula for the discriminant, \( \Delta = b^2 - 4ac \), to determine if there are real roots. Substitute the coefficients:\[ \Delta = 7^2 - 4 \times 4 \times (-12) = 49 + 192 = 241 \]

Step 4: Check for Real Roots

Since the discriminant \( \Delta = 241 \) is greater than zero, there are two distinct real roots.

Step 5: Apply the Quadratic Formula

Use the quadratic formula \( R = \frac{-b \pm \sqrt{\Delta}}{2a} \) to find the roots. First, calculate \( \sqrt{241} \approx 15.52 \).Substitute into the formula:\[ R_1 = \frac{-7 + 15.52}{8} \approx \frac{8.52}{8} = 1.07 \]\[ R_2 = \frac{-7 - 15.52}{8} \approx \frac{-22.52}{8} = -2.82 \]

Step 6: Round the Solutions

Round each solution to the nearest hundredth. Hence, the solutions are approximately \( R_1 = 1.07 \) and \( R_2 = -2.82 \).

Key Concepts

Discriminant in Quadratic Equations

The discriminant is a crucial part of solving quadratic equations. It helps us determine the nature and number of the roots (solutions) a quadratic equation might have. In a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is calculated using the formula:

• \( \Delta = b^2 - 4ac \)

After computing the discriminant, we can draw conclusions about the roots:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (or a "repeated" root).
• If \( \Delta < 0 \), the equation has no real roots, meaning the solutions are complex numbers.

In our example, we found \( \Delta = 241 \), which is greater than zero, indicating two real and distinct roots. This initial step is vital as it guides us on whether to calculate real roots or acknowledge that they do not exist.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of a quadratic equation. If you have your equation in the form \( ax^2 + bx + c = 0 \), the roots are given by:

• \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)

Here's how it works:1. Calculate the discriminant (\( \Delta \)).2. Plug the values of \( a \), \( b \), and \( \Delta \) into the formula.3. The ± sign indicates two possible values, resulting in two roots.
In our exercise, after computing \( \sqrt{241} \approx 15.52 \), we applied the quadratic formula:

• For the first root: \( R_1 = \frac{-7 + 15.52}{8} \approx 1.07 \)
• For the second root: \( R_2 = \frac{-7 - 15.52}{8} \approx -2.82 \)

The quadratic formula is reliable and can always be used to find the roots of any quadratic equation, which makes it indispensable in algebra.

Real Roots of Quadratics

Real roots in quadratic equations indicate where the parabola associated with the quadratic equation intersects the x-axis. Depending on the discriminant, these intersections might be:

• Two points (two distinct real roots).
• One point (one real root, touching the x-axis).
• No intersection (no real roots, which means the parabola does not touch the x-axis).

Real roots provide important information in various contexts, such as motion problems, optimization, and even graphical interpretations. In our given exercise, we determined two real roots because the discriminant was positive. Hence, the parabola intersects the x-axis at two distinct points.
Finding the real roots involves using both the discriminant to confirm their existence and the quadratic formula to pinpoint their values. This duality ensures we do not miss any real solutions and allows us to visualize the sum and product of the roots as properties of the quadratic coefficients.