Problem 71 - 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. $$4 x^{2}=6 x+3$$

Step-by-Step Solution

Answer

The equation has two irrational real solutions.

Step 1: Rewrite the Equation in Standard Form

First, rearrange the equation into the standard quadratic form, which is \( ax^2 + bx + c = 0 \). Start with the equation \( 4x^2 = 6x + 3 \) and move all terms to one side to get \( 4x^2 - 6x - 3 = 0 \).

Step 2: Identify the Coefficients

In the rewritten equation \( 4x^2 - 6x - 3 = 0 \), identify the coefficients: \( a = 4 \), \( b = -6 \), and \( c = -3 \).

Step 3: Compute the Discriminant

The discriminant \( \Delta \) is given by \( b^2 - 4ac \). Substitute the values of the coefficients into this formula to compute: \( \Delta = (-6)^2 - 4 \cdot 4 \cdot (-3) \). This simplifies to \( 36 + 48 = 84 \).

Step 4: Analyze the Discriminant

Since the discriminant \( \Delta = 84 \) is positive, the quadratic equation has two real solutions. 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 second degree. It looks like this: \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). The term \( x^2 \) indicates that the highest power of the variable \( x \) is 2, making it quadratic.

When dealing with quadratic equations, they can appear in different forms, and rearranging them into this standard form is typically the first step in solving or analyzing them. In the exercise, the equation \( 4x^2 = 6x + 3 \) was rearranged to \( 4x^2 - 6x - 3 = 0 \). This step allows you to easily identify the coefficients \( a = 4 \), \( b = -6 \), and \( c = -3 \).

Quadratic equations can be solved using various methods, including factoring, using the quadratic formula, or completing the square. However, before jumping into solving, understanding the nature of its solutions can be greatly aided by analyzing its discriminant.

Irrational Solutions

Irrational solutions in the context of quadratic equations refer to solutions that cannot be expressed as a simple fraction or exact decimal. They often involve square roots of numbers that are not perfect squares.

In the exercise, after computing the discriminant \( \Delta = 84 \), we determined whether the solutions are rational or irrational. Since 84 is not a perfect square, it indicates that the roots of the quadratic equation cannot be neatly expressed as fractions. Instead, they would be expressed in terms of square roots and are thus irrational.

Generally, when the discriminant \( \Delta > 0 \) and is not a perfect square, the solutions to the quadratic equation will be irrational. This is a common way to predict the type of solutions without actually solving the equation itself. Recognizing irrational solutions is crucial for understanding the nature of different equations and the methods needed for precise solutions.

Number of Real Solutions

The number of real solutions a quadratic equation has is directly related to its discriminant, \( \Delta \). The discriminant is calculated using the formula \( b^2 - 4ac \). After finding the value of the discriminant, you can predict the number of real solutions without actual computation of the roots.

The key scenarios are:

• If \( \Delta > 0 \), the quadratic equation has two distinct real solutions.
• If \( \Delta = 0 \), the quadratic equation has exactly one real solution, which is a double root.
• If \( \Delta < 0 \), the equation has no real solutions; the solutions are complex or imaginary.

In the exercise at hand, the discriminant was calculated to be 84, meaning there are two real solutions. By understanding these rules, you'll easily determine the quantity and nature of solutions for any quadratic equation by evaluating its discriminant.