Problem 90 - 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. $$3 x^{2}=4 x-5$$

Step-by-Step Solution

Answer

2 distinct non-real complex solutions.

Step 1: Rewrite the equation in standard form

Rewrite the given equation in the form of a quadratic equation, which is a x^{2} + b x + c = 0.Move all terms to one side: 3x^{2} - 4x + 5 = 0. Here, a = 3, b = -4, and c = 5.

Step 2: Identify the coefficients

From the standard form equation 3x^{2} - 4x + 5 = 0, the coefficients are:a = 3b = -4c = 5.

Step 3: Calculate the discriminant

The discriminant \Delta\ is calculated using the formula: \( \Delta = b^2 - 4ac \)Substitute the values of a, b, and c into the formula:\( \-4^2 - 4(3)(5) = 16 - 60 = -44 \)

Step 4: Determine the nature of the solutions

Use the value of the discriminant to determine the nature of the solutions:- If \( \Delta > 0 \), there are 2 distinct real solutions, and they are rational if \( \Delta \) is a perfect square, otherwise irrational.- If \( \Delta = 0 \), there is exactly one real solution.- If \( \Delta < 0 \), there are 2 distinct non-real complex solutions.Here, the discriminant \( \Delta \) is -44, which is less than 0. Therefore, there are 2 distinct non-real complex solutions.

Key Concepts

quadratic equation

A quadratic equation is a type of polynomial equation of the second degree. It generally has the form: \( ax^2 + bx + c = 0 \)where:

• \(a\) is the coefficient of \(x^2\), and it is always non-zero.
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

In the equation provided, 3\(x^2 = 4x - 5\), we first need to rewrite it in standard quadratic form. By moving all terms to one side, we get 3\(x^2 - 4x + 5 = 0\). Now we identify the coefficients as \(a = 3\), \(b = -4\), and \(c = 5\).The quadratic equation is fundamental in algebra and has a variety of methods for finding its solutions. These solutions could be real or complex numbers depending on the discriminant (more on this later).

• Quadratic equations can be solved by various methods like factoring, completing the square, or using the quadratic formula.

The solutions to quadratic equations are critical for many applications in science and engineering.

discriminant

The discriminant is a specific value calculated from the coefficients of a quadratic equation, and it helps us determine the nature and number of the solutions of the equation. The discriminant \( \Delta \) is given by the formula:\[ \Delta = b^2 - 4ac \]Here's what it tells us:

• If \( \Delta > 0 \), there are 2 distinct real solutions. If \( \Delta \) is a perfect square, the solutions are rational; otherwise, they are irrational.
• If \( \Delta = 0 \), there is exactly one real solution, and it is rational.
• If \( \Delta < 0 \), there are 2 distinct non-real complex solutions.

In our exercise, the coefficients are \(a = 3\), \(b = -4\), and \(c = 5\). Substituting these into the formula for the discriminant, we get:\[ \Delta = (-4)^2 - 4(3)(5) = 16 - 60 = -44 \]Since \( \Delta = -44 \) is less than zero, it indicates that our quadratic equation has two distinct non-real complex solutions.The discriminant is a highly useful concept in algebra and helps quickly understand the nature of solutions without actually solving the quadratic equation.

complex solutions

When the discriminant of a quadratic equation is less than zero (\( \Delta < 0 \)), the solutions are non-real complex numbers. Complex numbers take the form:\[ z = a + bi \]where:

• \(a\) represents the real part.
• \(bi\) represents the imaginary part.

Here, \(i\) is the imaginary unit and satisfies the equation \(i^2 = -1\). In our exercise, with \( \Delta = -44 \), this indicates that the solutions to the quadratic equation \(3x^2 - 4x + 5 = 0\) are complex.To better understand complex solutions, remember:

• Complex solutions always come in conjugate pairs, meaning if \(a + bi\) is a solution, then \(a - bi\) is also a solution.
• These solutions are not real numbers and cannot be plotted on the traditional number line but can be represented on the complex plane.

Complex solutions are significant in various fields, including engineering and physics, where they are used to solve problems involving oscillations, waves, and many other phenomena.