Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$4 x^{2}=6 x+3$$

Step-by-Step Solution

Answer

The quadratic equation has two distinct irrational real solutions.

Step 1: Convert to Standard Form

First, let's rewrite the given quadratic equation \(4x^2 = 6x + 3\) in the standard form \(ax^2 + bx + c = 0\). To do this, subtract \(6x\) and \(3\) from both sides to get \(4x^2 - 6x - 3 = 0\). Thus, we identify the coefficients as \(a = 4\), \(b = -6\), and \(c = -3\).

Step 2: Calculate the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\). Substituting the coefficients \(a = 4\), \(b = -6\), and \(c = -3\) into the formula, we find \(D = (-6)^2 - 4(4)(-3)\). This simplifies to \(D = 36 + 48 = 84\).

Step 3: Determine Number and Nature of Real Solutions

Since the discriminant \(D = 84\), which is positive, the quadratic equation has two distinct real solutions. However, because 84 is not a perfect square, these solutions are irrational.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The largest exponent of the variable \(x\) in a quadratic equation is 2. This is the factor that makes it a quadratic.
Quadratic equations arise in numerous mathematical and real-world contexts, such as projectile motion, area calculations, and more. They graph as a parabola in the coordinate plane, which opens upwards if \(a > 0\) and downwards if \(a < 0\).
To solve a quadratic equation, one might use several methods:

• Factoring
• Completing the square
• Quadratic formula
• Graphing

Each of these methods can be applied based on the equation's specific characteristics. Sometimes, especially when dealing with non-factoring or higher coefficients, the quadratic formula is a useful tool: \\( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \).
This formula not only helps find solutions but is tightly linked with the concept of the discriminant, which gives insight into the nature of these solutions.

Real Solutions

When discussing the solutions of quadratic equations, the term 'real solutions' refers to the roots of the equation that are real numbers. A real number is any value on the continuous number line that is not imaginary.
In the context of quadratic equations, the discriminant \(D\) from the formula \(b^2 - 4ac\) is crucial in determining not just the number of solutions but also their nature.
Here's how you can interpret the discriminant:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, which is a repeated root.
• If \(D < 0\), there are no real solutions, but two complex solutions.

Understanding the discriminant is fundamental when analyzing quadratic equations because it provides a quick assessment of how many solutions exist and whether they are real or complex.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction, meaning their decimal representation never ends or repeats. Classic examples include \(\pi\) and the square root of non-perfect squares like \(\sqrt{2}\).
In the study of quadratic equations, irrational numbers often appear when the discriminant \(D = b^2 - 4ac\) is positive but not a perfect square. This means when you plug \(D\) into the quadratic formula, \(\sqrt{D}\) results in an irrational number, leading to solutions that are not integers or simple fractions.
Irrational solutions are significant in many fields, including engineering and physics, as they often represent more complex phenomena or outcomes. The recognition and handling of these numbers are thus essential in advanced mathematical problem-solving and understanding real-world applications.