Problem 9 - Technical Mathematics with Calculus

Question

Find the roots of each quadratic by any of the methods shown in this section. Keep three significant digits. For some, use more than one method and compare results. Explicit Functions. $$49 x^{2}+21 x-5=0$$

Step-by-Step Solution

Answer

The roots of the equation \(49x^2+21x-5=0\) are approximately \(x \approx 0.102\) and \(x \approx -1.03\), using the quadratic formula rounded to three significant figures.

Step 1: Use the Quadratic Formula

The quadratic formula states that the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\) can be found using \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For the equation \(49x^2+21x-5=0\), identify \(a = 49\), \(b = 21\), and \(c = -5\). Plug these values into the quadratic formula to find the roots.

Step 2: Calculate the Discriminant

Compute the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots. Here, \(\Delta = 21^2 - 4 \cdot 49 \cdot (-5)\).

Step 3: Evaluate the Quadratic Formula

After calculating the discriminant, substitute \(a\), \(b\), and \(\Delta\) into the quadratic formula and solve for \(x\) to obtain the two roots.

Step 4: Simplify the Results

Simplify the results from the quadratic formula and round them to three significant digits.

Step 5: Verify with Another Method (Optional)

If required, verify the solutions by comparing them with the results obtained through another method, such as factoring, completing the square, or graphing (if possible). This step is optional and based on the specific directive of the exercise.

Key Concepts

Root Finding

Finding the roots of a quadratic equation is a fundamental skill in algebra. It involves determining the values of the variable, typically represented as 'x', that make the equation equal to zero. A quadratic equation has the general form of \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are coefficients.

There are several methods for root finding, including factoring, completing the square, graphing, and using the quadratic formula, which is a reliable method that works for any quadratic equation. When the quadratic equation cannot be easily factored, the quadratic formula is the go-to method for solutions. It's crucial to note that a quadratic equation may have two real roots, one real root, or two complex roots depending on the discriminant value, which we'll explore in the next section.

Discriminant

The discriminant is a significant component in the quadratic formula and plays a key role in determining the nature of the roots of a quadratic equation. It is represented by the symbol \( \Delta \) and is calculated as \( \Delta = b^2 - 4ac \) from the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

Nature of Roots

Based on the discriminant value, the nature of the roots can be determined as follows:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root (also known as a repeated or double root).
• If \( \Delta < 0 \), the equation has two complex roots.

Understanding the discriminant is crucial for predicting the type of roots without solving the equation.

Significant Digits

When solving mathematical problems, especially in scientific and engineering contexts, the concept of significant digits (often called significant figures) is essential. Significant digits help indicate the precision of an answer, reflecting the accuracy of the data provided.

Maintaining the correct number of significant digits in the solutions of a quadratic equation is important to ensure the reliability of the results. The rule of thumb is the number of significant digits in the answer should not exceed those in the given coefficients of the equation. For example, if the coefficients are provided with three significant figures, the roots should also be presented with three significant figures. This respects the precision of the initial measurements or given data, preventing the implication of a false sense of accuracy in the calculated values.