Problem 85 - Precalculus Student Solutions Manual 5th

Question

Evaluate the discriminant for each equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or non real complex. Do not solve the equation. $$3 x^{2}+5 x+2=0$$

Step-by-Step Solution

Answer

The discriminant is 1, so there are two distinct real and rational solutions.

Step 1: Identify coefficients

From the equation \(3x^2 + 5x + 2 = 0\), identify the coefficients as follows:\(a = 3\), \(b = 5\), \(c = 2\).

Step 2: Write the discriminant formula

Recall the formula for the discriminant of a quadratic equation, which is given by \(D = b^2 - 4ac\).

Step 3: Substitute the coefficients into the discriminant formula

Substitute \(a = 3\), \(b = 5\), and \(c = 2\) into the formula: \[D = 5^2 - 4(3)(2)\].

Step 4: Simplify the expression

Simplify to find the value of the discriminant: \[D = 25 - 24 = 1\].

Step 5: Analyze the discriminant

Since the discriminant \(D = 1\) is greater than zero and a perfect square, the equation has two distinct real and rational solutions.

Key Concepts

distinct solutions

When solving quadratic equations, the number of distinct solutions is determined by the discriminant, denoted as \(D\). The discriminant formula is given by \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients from the standard form of the quadratic equation \(ax^2 + bx + c = 0\). If \(D > 0\), the equation has two distinct solutions. These solutions will be real numbers.
If \(D = 0\), the equation has exactly one distinct solution, which is a repeated real root. If \(D < 0\), there are no real solutions; instead, there will be two distinct complex solutions.
Based on the provided exercise, we can see that the equation \(3x^2 + 5x + 2 = 0\) has a discriminant \(D = 1\), which is greater than zero. This tells us the equation has two distinct real solutions.

rational solutions

A solution to a quadratic equation is classified as rational or irrational based on the discriminant and the quadratic formula. A rational solution means that the solutions are numbers that can be expressed as fractions of integers. If the discriminant \(D\) is a perfect square (like 1, 4, 9, etc.), then the solutions will be rational.
On the other hand, if \(D\) is not a perfect square, the solutions will involve square roots of non-square numbers, leading to irrational solutions. In the exercise, the discriminant \(D = 1\) is a perfect square. Therefore, the solutions are rational.
This implies that the quadratic equation \(3x^2 + 5x + 2 = 0\) has two rational solutions.

quadratic formula

The quadratic formula is a tool used to find the solutions of any quadratic equation in the form \(ax^2 + bx + c = 0\). The formula is:

• \(x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a}\)

Here, \(b^2 - 4ac\) represents the discriminant \(D\). The plus-minus symbol \(\pm\) indicates that the quadratic equation will typically have two solutions:

• \(x = \frac{{-b + \sqrt{D}}}{2a}\)
• \(x = \frac{{-b - \sqrt{D}}}{2a}\)

The quadratic formula provides a systematic way to find solutions, whether they are distinct, rational, irrational, or complex. The provided exercise shows an application of the discriminant, but if we applied the quadratic formula to the coefficients \(a = 3\), \(b = 5\), and \(c = 2\), and knowing \(D = 1\), we would arrive at the rational solutions step-by-step.