Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$9 x^{2}-6 x+37=0$$

Step-by-Step Solution

Answer

The quadratic equation \(9x^{2}-6x+37=0\) has no real roots (solutions). The answers are complex numbers, verified graphically with a parabola that does not intersect the x-axis.

Step 1: Identify the coefficients

For the given equation \(9x^{2}-6x+37=0\), the coefficients are 'a' = 9, 'b' = -6, and 'c' = 37.

Step 2: Substitute the coefficients into the quadratic formula

Substitute the values of 'a', 'b', and 'c' into the quadratic formula: \(x= \frac{-(-6)\pm\sqrt{(-6)^{2}-4*(9)*(37)}}{2*(9)}\).

Step 3: Simplify

After calculations, we realize that under the square root there is a negative, meaning there is no real root to the equation. In this case, we would simplify to \(x= \frac{6\pm\sqrt{(-36)}}{18}\), which gives us two complex roots.

Step 4: Verify graphically

Use a graphing utility to plot the function \(f(x) = 9x^{2}-6x+37\). As this is an upward parabola with all positive real values, and the solutions to the equation are complex numbers, the parabola doesn't intersect the x-axis, which confirms that there are no real solutions to the equation.

Key Concepts

Complex Roots

When solving quadratic equations like \(9x^{2}-6x+37=0\text{)}, you may encounter a situation where the equation does not have real solutions. This occurs when the discriminant, the part under the square root of the quadratic formula (\text{b}^{2}-4\text{ac}\)), is negative. In our exercise, the discriminant is \( (-6)^{2}-4*(9)*(37)\), which simplifies to a negative number. This leads to the square root of a negative number, indicating that the solutions are complex numbers rather than real numbers.

Complex numbers have a real part and an imaginary part, typically expressed as \(a + bi\), where \(i\) represents the imaginary unit, defined as the square root of -1. In our equation, the complex roots take the form \(x=\frac{3\pm i\sqrt{147}}{9}\), asserting that the graph of this quadratic equation does not intersect the x-axis, which is a key feature of parabolas with real roots.

Graphing Utility

A graphing utility is an invaluable tool for visualizing the behavior of quadratic functions and confirming the nature of their roots. These digital tools allow students and mathematicians to plot equations and observe graph characteristics such as vertex, direction of opening, and intercepts. In this case, by inputting the function \(f(x) = 9x^{2}-6x+37\), the graphing utility would display an upward-opening parabola.

The visualization confirms what the quadratic formula has revealed: since the parabola does not intersect the x-axis, the function has no real zeros, and the roots are indeed complex. Using a graphing utility not only aids in the verification of algebraic solutions but also enhances conceptual understanding by making the abstract properties of quadratic equations more tangible and easier to grasp.

Parabola

The graph of any quadratic equation \(y=ax^{2}+bx+c\), where \(a \eq 0\), is known as a parabola. This curve has a distinct 'U' shape that can open upwards or downwards depending on the sign of the coefficient ‘a’. If 'a' is positive, as in \(9x^{2}-6x+37\), the parabola opens upward. Characteristics such as the vertex, axis of symmetry, and direction of opening are crucial for understanding the motion and intersections—or lack thereof—of the parabola.

In the absence of real roots, as is the case with our example equation, the parabola does not cross the x-axis, indicating that the vertex is above the x-axis and the parabola remains entirely in the upper half of the coordinate plane. Recognizing the orientation and position of a parabola is essential in predicting the nature of its roots and is key in the graphical understanding of quadratic functions.