Problem 13 - 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. $$3 x^{2}+5 x=7$$

Step-by-Step Solution

Answer

The roots of the quadratic equation \(3x^2 + 5x - 7 = 0\) are approximately \(x_1 \approx 0.868\) and \(x_2 \approx -2.53\).

Step 1: Rearrange the equation

First, rearrange the equation into standard quadratic form, which is \(ax^2 + bx + c = 0\). Subtract 7 from both sides of the given equation to get \(3x^2 + 5x - 7 = 0\).

Step 2: Apply the quadratic formula

Use the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) where \(a = 3\), \(b = 5\), and \(c = -7\).

Step 3: Calculate the discriminant

Calculate the discriminant \(\Delta = b^2 - 4ac\) which is \(\Delta = 5^2 - 4(3)(-7)\), so \(\Delta = 25 + 84 = 109\).

Step 4: Find the roots

Since \(\Delta > 0\), there are two real roots. Plug the values into the quadratic formula to get \(x = \frac{{-5 \pm \sqrt{109}}}{{6}}\).

Step 5: Simplify the expression

Simplify the expression to find the two roots: \(x_1 = \frac{{-5 + \sqrt{109}}}{{6}}\) and \(x_2 = \frac{{-5 - \sqrt{109}}}{{6}}\). Round both roots to three significant digits.

Step 6: Round the solutions

The roots rounded to three significant digits are \(x_1 \approx 0.868\) and \(x_2 \approx -2.53\).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental solution method for quadratic equations, which are equations of the second degree, typically written in the form of \( ax^2 + bx + c = 0 \). The formula is given by \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \), where \(a\), \(b\), and \(c\) are coefficients from the equation, with \(a\) not being zero. It allows for the calculation of the equation's roots, which can be real or complex numbers.

When applied correctly, it ensures that any quadratic equation can be solved, provided that one adheres to algebraic rules and performs calculations with care. If you stick close to the formula and carry out each step methodically, you will arrive at the correct roots of the quadratic equation.

Discriminant

The discriminant is a key component in the quadratic formula and is denoted by the Greek letter \(\Delta\). It's the part under the square root and is calculated as \(\Delta = b^2 - 4ac\). The value of the discriminant gives us crucial information about the nature of the roots:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root, also known as a repeated root.
• If \( \Delta < 0 \), the roots are complex and come in a conjugate pair.

Understanding the discriminant can not only tell us about the types of roots without even calculating them but also ensure we're prepared for the nature of the results we'll find using the quadratic formula.

Roots of Quadratic

The roots of a quadratic equation represent the values of \(x\) that make the equation \(ax^2 + bx + c = 0\) true. These can be found using several methods including factoring, completing the square, graphing, and the quadratic formula. When the discriminant \(\Delta\) is positive, we get two real roots; if it is zero, there is one real root; and a negative discriminant leads to complex roots.

In our example, after calculating and simplifying, the roots \(x_1 = \frac{{-5 + \sqrt{109}}}{{6}}\) and \(x_2 = \frac{{-5 - \sqrt{109}}}{{6}}\) are different and hence, two distinct real roots exist for the given equation.

Significant Digits

Significant digits (also referred to as significant figures) are a way of expressing precision in numerical answers. This concept is essential in science and engineering to avoid overstatement of the accuracy of a figure. The rule of three significant digits, as seen in the exercise example, means we express our answers in three digits that carry meaning contributing to its precision.

Therefore, rounding \( x_1 \) and \( x_2 \) to three significant digits gives us \( x_1 \approx 0.868 \) and \( x_2 \approx -2.53 \) respectively. This advised rounding ensures the balance between accuracy and practicality in presenting our computed roots.