Problem 73 - 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. $$\frac{2}{3} x^{2}+x=-1$$

Step-by-Step Solution

Answer

The quadratic equation \(\frac{2}{3} x^2 + x + 1 = 0\) does not have any real solutions, implying the corresponding quadratic function, \(f(x) = \frac{2}{3} x^2 + x + 1\), does not have any real zeros.

Step 1: Rearrange the Equation

First, rearrange the equation to the standard quadratic form: \(\frac{2}{3} x^2 + x + 1 = 0\) by adding 1 to both sides of the equation.

Step 2: Apply the quadratic formula

Apply the quadratic formula, \(x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}\), to find the solutions for \(x\). In our equation, \(a = \frac{2}{3}\), \(b = 1\), and \(c = 1\). Plugging these values in we get \[ x = \frac{-1 ± \sqrt{1^2 - 4*(\frac{2}{3}*1)}}{2*\frac{2}{3}} \] simplifying further we get \[ x = \frac{-1 ± \sqrt{1 - \frac{8}{3}}}{\frac{4}{3}} \] But the term inside the square root, \(\frac{1 - \frac{8}{3}}\), is less than zero, hence no real solutions exist for this equation.

Step 3: Relate solutions to the zeros of the quadratic function

The solutions found are the zeros of the corresponding quadratic function, \(f(x)=\frac{2}{3} x^2 + x + 1\). But, in this case, since no real solutions exist, it means the graph of the function does not intersect the x-axis. Thus, there are no real zeros for this function.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding solutions to quadratic equations. Quadratic equations are typically written in the form of \( ax^2 + bx + c = 0 \). Solving these equations involves finding the values of \( x \) which satisfy the equation. To do this, we use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a \), \( b \), and \( c \) are the coefficients of the terms of the equation.

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

Using the quadratic formula involves substituting these coefficients into the formula to determine the values of \( x \). The part under the square root, \( b^2 - 4ac \), is known as the discriminant. It helps us determine the nature of the roots. If the discriminant is positive, there are two distinct real roots. If it is zero, there is exactly one real root. If it is negative, the roots are complex numbers. This was the case in the exercise where we had a negative discriminant indicating no real solutions exist.

Complex Numbers

When dealing with quadratic equations, especially when the discriminant is negative, complex numbers come into play. Complex numbers extend our understanding of numbers beyond the real number line. They are written in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \). In situations where the quadratic formula results in a negative discriminant, the solutions are not real but complex. This means instead of finding real \( x \)-values, the solutions will involve complex numbers. This can often happen in quadratic equations like the one in the exercise, where the discriminant \( 1 - \frac{8}{3} \) was negative, resulting in complex solutions. Solutions of this type do not appear on the traditional x-y graph but are still important in many areas of mathematics.

Zeros of a Function

The zeros of a function are the values of \( x \) that make the function equal to zero. For a quadratic function \( f(x) = ax^2 + bx + c \), finding the zeros involves solving the equation \( ax^2 + bx + c = 0 \). Essentially, these zeros are the solutions of the quadratic equation and represent where the graph of the polynomial intersects the x-axis. However, if the discriminant is negative, as in our example, the graph does not intersect the x-axis at any point. Thus, there are no real zeros. This means that the solutions - or zeros - are only theoretical and consist of complex numbers. Understanding zeros helps in graphing and analyzing the behavior of functions. Even if no real zeros exist, the complex zeros still provide significant insight into the function's properties and behavior.