Problem 68 - 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. $$7 x^{2}=-x-1$$

Step-by-Step Solution

Answer

The solutions for the given quadratic equation are \(x = -\frac{1}{14} + \frac{i\sqrt{27}}{14}\) and \(x = -\frac{1}{14} - \frac{i\sqrt{27}}{14}\).

Step 1: Rearrange the Equation

First, rearrange the equation in standard form by moving all of the terms to one side: \(7x^2 + x + 1 = 0\).

Step 2: Use Quadratic Formula

Next, apply the quadratic formula to find solutions: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 7\), \(b = 1\) and \(c = 1\).

Step 3: Substituting the values

Substitute the values of \(a\), \(b\), and \(c\) that were found in step 2 into the formula which gives: \(x = \frac{-1 \pm \sqrt{(1)^2 - 4*7*1}}{2*7}\).

Step 4: Calculate Inside the Square Root

Calculate the value inside the square root (the discriminant): \(1-28 = -27\).

Step 5: Evaluate the Square Root

Evaluate the square root of -27. Since this is negative, the solutions will be complex numbers. Therefore, we have \(x = \frac{-1 \pm i\sqrt{27}}{14}\).

Step 6: Simplify the Solution

Simplify the solution to get the final answer. Dividing \(-1\) and \(\pm i\sqrt{27}\) by \(14\) respectively gives \(x = -\frac{1}{14} \pm \frac{i\sqrt{27}}{14}\).

Key Concepts

Quadratic Formula

The quadratic formula is a tool used to solve quadratic equations. These are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The formula is an easy-to-remember tool that provides solutions for any quadratic equation, making it particularly useful when factoring is difficult or impossible. The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula computes the values for \(x\) that satisfy the equation (make the equation true).

• \(b^2 - 4ac\) is called the discriminant and it plays a crucial role in determining the nature of solutions.
• "\(\pm\)" indicates that there are usually two possible solutions.

When applied, the quadratic formula requires identifying the coefficients \(a\), \(b\), and \(c\) from the equation and substituting them into the formula. Then, work through the arithmetic to find your solutions.

Complex Numbers

Complex numbers occur when dealing with square roots of negative numbers. In the context of the quadratic formula, if the discriminant \(b^2 - 4ac\) is negative, this indicates that the solutions will be complex numbers. Complex numbers are in the form \(a + bi\), where \(i\) is the imaginary unit, equivalent to \(\sqrt{-1}\).

• When calculating \(\sqrt{\text{negative number}}\), the result is a complex number.
• In our exercise, the discriminant was \(-27\), which led to solutions like \(-\frac{1}{14} \pm \frac{i \sqrt{27}}{14}\).

Complex solutions often appear in conjugate pairs—if one solution is \(x = a + bi\), another might be \(x = a - bi\). This symmetrical nature of complex numbers helps maintain the balance of quadratic solutions.

Zeros of a Function

The zeros of a function are the values of \(x\) that make the function equal to zero. In a quadratic function, these zeros are the solutions to the quadratic equation.

• Finding the zeros of a function provides valuable insights into the behavior of the function on a graph.
• For a quadratic equation like \(7x^2 + x + 1 = 0\), the zeros are the solutions found using the quadratic formula.

In a real-world scenario, zeros can represent times when an object hits the ground, or when a business breaks even—any situation where we want to know when two quantities are equal or balance each other out.
Applying this to our original equation, the zeros we found, \(-\frac{1}{14} \pm \frac{i \sqrt{27}}{14}\), indicate the points on the graph where the quadratic wouldn't realistically intersect the \(x\)-axis due to being complex. This highlights the intersection's non-visual, *complex* nature in place of where real solutions would usually show such crossings.