Problem 17 - 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. Implicit Functions. $$6 x-300=205-3 x^{2}$$

Step-by-Step Solution

Answer

The roots of the equation \(3x^2 + 6x - 505 = 0\) are approximately \(x_1 \approx 8.67\) and \(x_2 \approx -19.4\).

Step 1: Rearrange the Equation

Rearrange the given equation into the standard quadratic form by moving all terms to one side: \(3x^2 + 6x - (205 + 300) = 0\), which simplifies to \(3x^2 + 6x - 505 = 0\).

Step 2: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots. Here, \(a = 3\), \(b = 6\), and \(c = -505\).

Step 3: Calculate the Discriminant

Calculate the discriminant \(\Delta = b^2 - 4ac\), which is needed for the quadratic formula. Substituting the values gives \(\Delta = 6^2 - 4(3)(-505)\).

Step 4: Find the Roots

Find the roots using the values of \(a\), \(b\), and \(\Delta\) in the quadratic formula to calculate the two possible solutions for \(x\).

Step 5: Simplify the Roots

Simplify the roots to obtain the solutions in decimal form and round to three significant digits if necessary.

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool that provides a method for solving quadratic equations of the form \(ax^2 + bx + c = 0\). This formula states that the solutions, or roots, of the quadratic equation are given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The \(\pm\) symbol indicates that there are typically two solutions: one where you add the square root term, and another where you subtract it.

In the provided exercise, we applied the quadratic formula to solve the equation \(3x^2 + 6x - 505 = 0\). Using the coefficients \(a = 3\), \(b = 6\), and \(c = -505\), we plugged these values into the formula to find the roots. The quadratic formula is especially useful when factoring is difficult or impossible, and it always works provided the equation is in the correct quadratic form and \(a eq 0\).

Discriminant

The Discriminant is a key part of the quadratic formula that determines the nature and number of roots of a quadratic equation. It is represented by the symbol \(\Delta\) and is given by the expression \(\Delta = b^2 - 4ac\).

The value of the discriminant can tell us three things:

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

In the exercise, we calculated the discriminant by inserting the values of \(a\), \(b\), and \(c\) into the discriminant formula, resulting in \(\Delta = 6^2 - 4(3)(-505)\). We then used this value to determine the number and type of solutions the quadratic equation has.

Roots of Quadratic Equations

The roots of quadratic equations are the values of \(x\) that make the equation \(ax^2 + bx + c = 0\) true. These roots can be real or complex and there can be either one or two of them depending on the discriminant's value.

After computing the discriminant in the quadratic formula, we are ready to find the roots. For our equation \(3x^2 + 6x - 505 = 0\), we had a positive discriminant, indicating two distinct real roots. By substituting the known values of \(a\), \(b\), and \(\Delta\) into the quadratic formula, we obtained the roots. The final step involved simplifying these roots to the required number of significant digits as specified in the problem statement.

Understanding the roots of quadratic equations is vital as they often represent key points in various applications such as physics, engineering, and economics. By learning how to apply the quadratic formula and interpret the discriminant, students can solve a wide range of problems with confidence.