Problem 19 - 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. $$-3 x^{2}+9 x-5=0$$

Step-by-Step Solution

Answer

The solutions are approximately 0.74 and 2.26.

Step 1: Identify the Equation Format

The given equation is a quadratic equation in the standard form: \[-3x^2 + 9x - 5 = 0\] where \(a = -3\), \(b = 9\), and \(c = -5\).

Step 2: Calculate the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula: \[D = b^2 - 4ac\] Substitute \(a = -3\), \(b = 9\), and \(c = -5\) into the formula:\[D = 9^2 - 4(-3)(-5) = 81 - 60 = 21\] Since \(D > 0\), the equation has two real roots.

Step 3: Use the Quadratic Formula

The quadratic formula to find the roots is: \[x = \frac{-b \pm \sqrt{D}}{2a}\] Substitute \(b = 9\), \(D = 21\), and \(a = -3\) into the formula:\[x = \frac{-9 \pm \sqrt{21}}{-6}\]

Step 4: Calculate the Roots Using a Calculator

First, calculate \(\sqrt{21}\) using a calculator to get approximately 4.58. Next, find the two roots:\[x_1 = \frac{-9 + 4.58}{-6} \approx \frac{-4.42}{-6} \approx 0.74\]\[x_2 = \frac{-9 - 4.58}{-6} \approx \frac{-13.58}{-6} \approx 2.26\]

Step 5: Round to Nearest Hundredth

The roots calculated are approximately 0.7367 and 2.2633. When rounded to the nearest hundredth, the roots are 0.74 and 2.26, respectively.

Key Concepts

Discriminant

In the world of quadratic equations, the discriminant is a powerful tool. It helps determine the nature of the roots without actually solving the equation. The discriminant is simply a number that can be calculated directly from the coefficients of a quadratic equation in the form: \[ ax^2 + bx + c = 0 \]The formula to find the discriminant is:\[ D = b^2 - 4ac \]Here's why the discriminant matters:

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root (which is a repeated root, also known as a double root).
• If \( D < 0 \), the equation has no real roots, meaning the solutions are complex numbers.

In our exercise, we calculated the discriminant as \( D = 21 \). Since \( D > 0 \), we know that our quadratic equation has two real roots.

Quadratic Formula

The quadratic formula provides a straightforward method to find solutions, or roots, for any quadratic equation. The formula works as follows:\[ x = \frac{-b \pm \sqrt{D}}{2a} \]Where:

• \( b \) is the coefficient of the linear term \( x \)
• \( D \) is the discriminant calculated through \( D = b^2 - 4ac \)
• \( a \) is the coefficient of \( x^2 \)

What makes the quadratic formula especially useful is its ability to deliver both roots in one swoop. The "\( \pm \)" symbol means you'll perform the operation twice:

• Once as "\( -b + \sqrt{D} \)"
• Once as "\( -b - \sqrt{D} \)"

In our exercise, substituting \( a = -3 \), \( b = 9 \), and \( D = 21 \) into the formula gives us the two solutions or roots for the equation.

Real Roots

Real roots refer to the solutions of the quadratic equation that are real numbers. These no-nonsense solutions are the values of \( x \) that satisfy the equation, and they can be plotted on a real number line.Let's break down how we obtained the real roots in our example:

• We had \( x_1 = \frac{-9 + 4.58}{-6} \) and \( x_2 = \frac{-9 - 4.58}{-6} \).
• Using a calculator to simplify these gave us the approximate roots: \( x_1 = 0.74 \) and \( x_2 = 2.26 \).

These values are the points where the parabola, represented by the quadratic equation, intersects the x-axis on a graph. Since we detected \( D > 0 \) in our discriminant, it confirmed the existence of these real roots, rather than complex ones. These roots are practical as they can represent possible solutions in real-world problems, such as determining time, distance, or other measurable quantities. Understanding when and how to find real roots helps shed light on the behavior of quadratic equations beyond just numbers!.