Problem 18 - 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. $$3 x^{2}-25 x=5 x-73$$

Step-by-Step Solution

Answer

The roots of the quadratic equation are approximately \(x_1 \approx 5.82\) and \(x_2 \approx 4.18\), to three significant digits.

Step 1: Move All Terms to One Side

To find the roots of the quadratic equation, we need to have the equation in the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). So, let's move all terms to one side to get the equation into standard form: \[3 x^{2}-25 x - (5 x-73) = 0\]and simplify it.\[3 x^{2}-25 x - 5 x + 73 = 0\]Now combine like terms.\[3 x^{2}-30 x + 73 = 0\]

Step 2: Choose a Method to Find the Roots

We can solve the quadratic equation \(3 x^{2}-30 x + 73 = 0\) using various methods like factoring, completing the square, the quadratic formula, or graphically. Factoring might not be straightforward due to the coefficients and constant, so let's use the quadratic formula.

Step 3: Use the Quadratic Formula

The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]For the equation \(3 x^{2}-30 x + 73 = 0\), the coefficients are: \(a = 3\), \(b = -30\), and \(c = 73\). Plugging these values into the quadratic formula gives us:\[x = \frac{-(-30) \pm \sqrt{(-30)^2-4(3)(73)}}{2(3)}\]Simplify inside the square root and calculate the values of x.

Step 4: Calculate the Discriminant

Before further computation, let's find the discriminant (\(b^2-4ac\)) to determine the nature of the roots.\[\Delta = (-30)^2 - 4(3)(73)\]\[\Delta = 900 - 876\]\[\Delta = 24\]Since the discriminant is positive, there will be two distinct real roots.

Step 5: Solve for x

Now, compute the values of x:\[x = \frac{30 \pm \sqrt{24}}{6}\]Since we need to keep three significant digits, let's simplify further.\[x_{1,2} = 5 \pm \frac{\sqrt{24}}{6}\]And now calculate the numerical values of the two roots:\[x_1 = 5 + \frac{\sqrt{24}}{6}\]\[x_2 = 5 - \frac{\sqrt{24}}{6}\]

Key Concepts

Solving Quadratic Equations

Quadratic equations are a staple in algebra and come in the form, \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(x\) represents the variable. Solving these equations means finding the values of \(x\) that make the equation true. There are various methods to solve quadratic equations, including factoring, completing the square, using the quadratic formula, or graphing.

For equations that are difficult to factor, the quadratic formula is a reliable tool. However, if the quadratic is easily factorable, this method can provide a quick solution. Graphical methods can also offer a visual understanding of where the function intersects the x-axis. A key step in solving quadratics is to ensure the equation is in standard form, which often involves combining like terms and moving all terms to one side of the equation—as we did in the initial steps of our example problem.

Quadratic Formula

When quadratic equations cannot be solved by factoring or simple methods, the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) is a fail-safe option. It is derived from the process of completing the square and gives us the roots of any quadratic equation. By simply identifying the coefficients \(a\), \(b\), and \(c\) and substituting them into the formula, we can find the solutions for \(x\).

Application of the Quadratic Formula

As demonstrated in our example, we applied the quadratic formula by plugging in the values of \(a = 3\), \(b = -30\), and \(c = 73\) from the equation \(3 x^{2}-30 x + 73 = 0\) to find the roots. The formula is especially useful for equations with coefficients that make factoring complex.

Discriminant of a Quadratic

The discriminant, often denoted as \(\Delta\), is a component derived from the quadratic formula and is given by the expression \(b^2 - 4ac\). It provides crucial information about the nature and number of roots of a quadratic equation without solving it completely.

The discriminant can result in three scenarios:

• If \(\Delta > 0\), there are 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; instead, there are two complex roots.

In our example, \(\Delta = 24\), a positive number, indicating two distinct real roots, which aligns with the roots we later calculated.

Significant Digits in Quadratics

When solving quadratic equations, the accuracy of the solution can be crucial, especially in scientific and engineering contexts. Significant digits ensure this precision by limiting the number of digits to communicate the precision of the calculation. In our exercise, we are asked to round the roots to three significant digits, striking a balance between accuracy and manageability.

It is important to apply significant digits at the end of the calculation to avoid rounding errors during intermediate steps. As shown in our example, we initially worked with exact values, and only after calculating both roots did we round them to three significant digits, which ensures that we have a precise and reliable solution.