Problem 74 - A Graphical Approach to College Algebra

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. $$3 x^{2}=4 x-5$$

Step-by-Step Solution

Answer

The equation has no real solutions because the discriminant is negative.

Step 1: Rewrite the Equation in Standard Form

The given equation is \(3x^2 = 4x - 5\). First, rearrange it into the standard quadratic form \(ax^2 + bx + c = 0\). Move all terms to one side to get \(3x^2 - 4x + 5 = 0\). Here, \(a = 3\), \(b = -4\), and \(c = 5\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(\Delta = b^2 - 4ac\). Substitute the values \(a = 3\), \(b = -4\), and \(c = 5\) into the discriminant formula: \(\Delta = (-4)^2 - 4 \times 3 \times 5\).

Step 3: Simplify the Discriminant Expression

Now simplify the expression: \(\Delta = 16 - 60 = -44\).

Step 4: Interpret the Discriminant

Since the discriminant \(\Delta = -44\) is less than zero, the quadratic equation has no real solutions. Instead, it has two complex solutions.

Key Concepts

Real Solutions

A real solution refers to the values of variables that satisfy the equation and can be represented on the real number line. In the context of quadratic equations, a real solution implies that the curve of a quadratic function intersects the x-axis at one or more points.
When solving quadratic equations, the discriminant, which is the part of the quadratic formula under the square root sign, determines the nature of the roots.

• If the discriminant is positive, the equation has two distinct real solutions.
• If it is zero, the equation has exactly one real solution, sometimes called a repeated or double root.
• However, if the discriminant is negative, the equation has no real solutions.

In our original exercise, the discriminant was found to be negative, indicating the absence of real roots.

Quadratic Equation

A quadratic equation is any equation that can be rearranged in the form of \[ax^2 + bx + c = 0\] where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). This equation is a second-degree polynomial, the highest exponent on the variable \(x\) being 2.

Quadratic equations graph as parabolas on the coordinate plane, and the solutions of the equation represent the points where the parabola intersects the x-axis.

These intersections are also known as the roots or zeros of the equation.

• Quadratic equations can have two, one, or no real solutions.
• The nature of these solutions is determined by the discriminant.

In the exercise, we rearranged the given equation \(3x^2 = 4x - 5\) into the quadratic form \(3x^2 - 4x + 5 = 0\), which allowed us to evaluate its discriminant.

Complex Solutions

When the discriminant of a quadratic equation is negative, the equation does not intersect the x-axis, meaning no real solutions exist. Instead, the solutions are complex numbers. Complex numbers include an imaginary unit, denoted as \(i\), defined as \(i = \sqrt{-1}\).

If you have a negative discriminant, it implies the presence of complex solutions, which come in conjugate pairs. This means if \(a + bi\) is a solution, \(a - bi\) is also a solution.

In our exercise example, the discriminant calculated was \(-44\), thus confirming the quadratic equation \(3x^2 - 4x + 5 = 0\) has no real solutions, but instead, two complex solutions.

• These solutions cannot be plotted on the real number line, but can be visually represented on the complex plane.

This is significant, particularly in fields that require comprehensive solutions beyond real numbers.