Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to determine the number of real solutions of the quadratic equation. $$9+2.4 x-8.3 x^{2}=0$$

Step-by-Step Solution

Answer

The given quadratic equation \(9+2.4x-8.3x^2=0\) has two real solutions.

Step 1: Rewrite the equation

First, rewrite the quadratic equation in the standard form, \(ax^2 + bx + c = 0\). The given equation can be rewritten as \(-8.3x^2 + 2.4x + 9 = 0\) where \(a = -8.3\), \(b = 2.4\) and \(c = 9\).

Step 2: Calculate the discriminant

Next, calculate the discriminant using the formula \(b^2 - 4ac\). This equals \(2.4^2 - 4*(-8.3)*9 = 313.92.\)

Step 3: Determine the number of real solutions

Since the discriminant is greater than 0 (\(313.92 > 0\)), there are two real solutions to the equation.

Step 4: Plot the equation

To confirm this result visually, plot the equation \(-8.3x^2 + 2.4x + 9 = 0\) using a graphing utility. The graph will be a downward facing parabola, intercepting the x-axis at the two real solutions.

Key Concepts

Discriminant of a Quadratic Equation

When you come across a quadratic equation, one of your first questions might be how many real solutions it has. The discriminant of a quadratic equation is your go-to tool for this. It is part of the quadratic formula and is found using the formula \(b^2 - 4ac\). This can tell you whether your solutions are real numbers, but also how many of them there are. A positive discriminant indicates that there are two different real solutions, which is evident in our exercise where the discriminant is 313.92, well above zero. A discriminant of zero means there is only one real solution, known as a repeated or double root. Lastly, a negative discriminant implies that no real solutions exist, and the equation has two complex solutions instead.

Understanding the discriminant gives you a quick way to gauge the nature of the solutions without needing to calculate them fully, saving precious time especially during exams.

Standard Form of a Quadratic Equation

Knowing the right form of a quadratic equation is crucial to solving it correctly. The standard form is given by \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. This form is intuitive and crucial because it sets the stage for most methods of finding the roots of the equation, including factoring, using the quadratic formula, or completing the square. In the example exercise, rewriting the equation to \( -8.3x^2 + 2.4x + 9 = 0 \) is an essential first step that organizes the coefficients in a familiar pattern. Knowing the standard form also helps in identifying the 'a' term which tells us the direction the parabola will open when graphed – if \(a>0\), it opens upwards and if \(a<0\), like in our example, it opens downwards.

Graphing Utility Usage in Algebra

Graphing utilities have become invaluable in the study of algebra. They turn abstract equations into visual graphs, assisting you to better understand the behavior and the solutions of the equations. When you plot an equation like \( -8.3x^2 + 2.4x + 9 = 0 \) on a graphing calculator or software, it provides a visual representation, in this case, a parabola. Observing where the parabola crosses the x-axis gives you the real solutions of the equation.

For students and educators alike, graphing utilities provide an immediate visual confirmation and can be used to check the accuracy of algebraic solutions. They are particularly helpful in understanding the concepts of maxima and minima, intercepts, and the general shape of the graph. With the emergence of various online graphing tools, accessing a graphing utility is often only a click away, making it an accessible and highly useful learning aid in algebra.