Problem 66 - Precalculus : Building Concepts and Connections

Question

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$-7 x^{2}+2 x-1=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(-7x^{2} + 2x - 1 = 0\), which also are zeros of the concerned quadratic function, are \( x = \frac{1}{7} + \frac{i \sqrt{6}}{7}\) and \( x = \frac{1}{7} - \frac{i \sqrt{6}}{7} \).

Step 1: Understanding The Quadratic Formula

The quadratic formula is given by \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\] This formula provides the solution for the roots of any quadratic equation, where a, b, and c are coefficients from the general form of the quadratic equation (\(ax^{2} + bx + c = 0\)). In the task's equation, \(a = -7, b = 2, c = -1\).

Step 2: Plugging Coefficients Into The Quadratic Formula

Start solving the quadratic equation (-7x² + 2x - 1 = 0) by substituting \(-7\) for \(a\), \(2\) for \(b\), and \(-1\) for \(c\) in the formula. Thus, the equation becomes: \[x = \frac{-2 \pm \sqrt{(2^{2}) - 4*(-7)*(-1)}}{2*(-7)}\]

Step 3: Simplifying The Equation

Simplify the equation further: \[x = \frac{-2 \pm \sqrt{4 - 28}}{-14}\] \n After further simplification: \[x = \frac{-2 \pm \sqrt{-24}}{-14}\]

Step 4: Turning The Solution Into Complex Numbers

Because the value under the square root is negative (\(-24\)), the solutions to the quadratic equation are complex numbers (including an imaginary part). This is recognized by rewriting \(\sqrt{-24}\) as \(2i\sqrt{6}\) as the square root of \(-1\) is \(i\). Thus, the equation becomes: \[x = \frac{-2 \pm 2i \sqrt{6}}{-14}\]

Step 5: Final Solution

Simplify to the smallest form: \[ x = \frac{1}{7} \mp \frac{i \sqrt{6}}{7}\] So the solutions to the quadratic equation are \( x = \frac{1}{7} + \frac{i \sqrt{6}}{7}\) and \( x = \frac{1}{7} - \frac{i \sqrt{6}}{7}\). Those are complex numbers and represent the zeros of the corresponding quadratic function \(-7x^{2} + 2x - 1\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find solutions, or roots, of quadratic equations. If you have an equation in the form of \(ax^2 + bx + c = 0\), then the quadratic formula can be expressed as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula helps resolve any quadratic equation by simply plugging in the values of the coefficients \(a\), \(b\), and \(c\). For instance, in the equation \(-7x^2 + 2x - 1 = 0\), you assign \(a = -7\), \(b = 2\), and \(c = -1\). By substituting these values into the formula, you can begin to find the values of \(x\) that satisfy the equation. The intriguing part of the quadratic formula lies in the discriminant, \(b^2 - 4ac\), which determines the nature of the roots. If the discriminant is positive, you'll find two distinct real roots. However, if it's negative, as in this example, the roots will be complex.

Complex Numbers

Complex numbers play a crucial role when dealing with quadratic equations that result in a negative discriminant. A complex number is of the form \(a + bi\), where \(i\) is the imaginary unit satisfying \(i^2 = -1\). In our example, the quadratic equation resulted in \(\sqrt{-24}\), which is a negative number under the square root—something that can't be simplified into real numbers.Here's where complex numbers come into play: you rewrite \(\sqrt{-24}\) as \(2i\sqrt{6}\). This incorporation of \(i\) allows us to work with these otherwise unsolvable equations. Consequently, the solutions we found, \(x = \frac{1}{7} + \frac{i \sqrt{6}}{7}\) and \(x = \frac{1}{7} - \frac{i \sqrt{6}}{7}\), are complex numbers.

• Complex solutions show the relationship between quadratic equations and further branches of mathematics.
• They not only offer a solution but broaden the understanding of equations beyond real numbers.

Zeros of a Function

The zeros of a function, often referred to as roots, are the points where the graph of the function crosses the x-axis. For a quadratic function, specifically in the form \(f(x) = ax^2 + bx + c\), finding the zeros is akin to solving the equation \(ax^2 + bx + c = 0\). In simpler terms, it's finding the \(x\)-values that make the function \(f(x) = 0\).In our quadratic function \(-7x^2 + 2x - 1\), the zeros were determined using the quadratic formula to find \(x = \frac{1}{7} + \frac{i \sqrt{6}}{7}\) and \(x = \frac{1}{7} - \frac{i \sqrt{6}}{7}\). These solutions highlight that the function does not actually cross the x-axis in the realm of real numbers. Instead, they are located in the complex plane, indicating no real intersection with the x-axis on a standard 2D graph. This demonstrates how complex numbers offer insight into solving equations that would otherwise have no real solutions.