Problem 22 - 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. \(6 w-15=3 w^{2}\)

Step-by-Step Solution

Answer

There are no real roots for the equation as the discriminant is negative.

Step 1: Rearrange the Equation

Start by rearranging the equation into a standard quadratic form. Move all terms to one side resulting in: \[3w^2 - 6w + 15 = 0\]

Step 2: Calculate the Discriminant

To determine the number of real roots, calculate the discriminant using the formula \(b^2 - 4ac\). Here, \(a = 3\), \(b = -6\), and \(c = 15\). Substitute these values into the formula: \[(-6)^2 - 4(3)(15) = 36 - 180 = -144\]

Step 3: Interpret the Discriminant

Since the discriminant \(-144\) is less than zero, this indicates that there are no real roots for the equation.

Key Concepts

Understanding the Discriminant

The discriminant is a crucial component when working with quadratic equations. It can help us determine the number and type of roots, or solutions, an equation has. The discriminant is calculated using the formula:

• For a quadratic equation in the form of \(ax^2 + bx + c = 0\), the discriminant is given by \(b^2 - 4ac\).

The key lies in interpreting the value:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (also known as a repeated root).
• If it is negative, as seen in the original exercise with a discriminant of -144, there are no real roots.

Understanding the discriminant helps in predicting whether the parabolic graph of the quadratic equation will intersect the x-axis (real roots) or not at all (no real roots).
It makes the often complex nature of quadratic equations a bit easier to visualize and solve.

What Are Real Roots?

Real roots are the solutions to a quadratic equation that are real numbers. These represent the x-values at which the graph of the quadratic equation intersects the x-axis. Real roots can be distinct, meaning two different solutions, or they can be the same, which means the graph just touches the x-axis at one point.
Two scenarios can occur for real roots:

• When the discriminant is greater than zero, the quadratic equation has two distinct real roots. This means the parabola will cut through the x-axis at two points.
• When the discriminant is zero, the equation has one real root, indicating that the vertex of the parabola just touches the x-axis.

In the original exercise, since the discriminant was -144, there are no real roots.
"No real roots" means the graph does not touch or intersect the x-axis, implying complex or imaginary solutions. Understanding real roots and their connection to the discriminant is essential for graphically interpreting quadratic equations.

Applying the Quadratic Formula

The quadratic formula is a powerful tool that provides a direct way to find the roots of any quadratic equation in the form \(ax^2 + bx + c = 0\). It is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula incorporates the discriminant, \(b^2 - 4ac\), to determine the value under the square root.
The discriminant within the formula can help us quickly decide whether "real" solutions exist, and if they do, how many there are.

• If the under the square root is positive, we compute two values: \(\frac{-b + \sqrt{b^2 - 4ac}}{2a}\) and \(\frac{-b - \sqrt{b^2 - 4ac}}{2a}\), signifying two distinct real roots.
• If it's zero, we only compute one, indicating a repeated real root.
• If it's negative, as in the exercise, the formula yields no real solutions, resulting in imaginary values.

Using the quadratic formula is often faster and simpler, especially with a calculator, since it solves the problem in a direct, straightforward manner.

Using a Calculator in Mathematics

Calculators can be indispensable when solving mathematical problems, including quadratic equations. They swiftly handle complex calculations that might otherwise be time-consuming and error-prone when done by hand. Using a calculator, students can accurately compute the discriminant or apply the quadratic formula without manual errors.
In relation to the specific exercise, a calculator efficiently handles challenging calculations like \((-6)^2 - 4(3)(15) = -144\). This can help you quickly determine that there are no real roots.
When using calculators, consider these tips:

• Always double-check inputs to prevent mistakes, especially with negative values.
• Use parentheses to correctly maintain order of operations.
• Utilize the quadratic equation solvers available on many scientific calculators as a shortcut.

Mastering calculator usage not only enhances accuracy but also boosts confidence in tackling a wide range of mathematical problems, which takes the focus away from basic computation and frees you to concentrate on problem-solving and analysis.