Problem 59 For each of the following, find ... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 59

For each of the following, find the discriminant, $b^{2}-4 a c$ and then determine whether one real-number solution, two different real-number solutions, or two different imaginary-number solutions exist. $$x^{2}+3 x+4=0$$

Short Answer

Expert verified

The discriminant is -7, so there are two different imaginary-number solutions.

Step by step solution

- Identify coefficients

First, identify the coefficients from the quadratic equation in the form of $ax^2 + bx + c = 0$. For the given equation:$a = 1$, $b = 3$, and $c = 4$.

- Calculate the Discriminant

Use the formula for the discriminant, $b^2 - 4ac$. Substituting the identified coefficients:$b = 3$, $a = 1$, and $c = 4$,we get: $3^2 - 4(1)(4)$$= 9 - 16$$= -7$.

- Determine the Nature of the Solutions

Based on the value of the discriminant:- If the discriminant is greater than $0$, there are two different real-number solutions.- If the discriminant is equal to $0$, there is one real-number solution.- If the discriminant is less than $0$, there are two different imaginary-number solutions.Since the discriminant is $-7$, which is less than $0$, this equation has two different imaginary-number solutions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic equation

A quadratic equation is a polynomial equation of the form $ax^2 + bx + c = 0$, where *a*, *b*, and *c* are coefficients. The highest power of the variable $x$ is 2. This type of equation often appears in various fields such as physics, engineering, and economics.
To solve a quadratic equation, we typically use the quadratic formula: $x = \frac{{-b \, \pm \, \sqrt{{b^2 - 4ac}}}}{{2a}}$. This formula provides the solutions for $x$ depending on the value of the discriminant $b^2 - 4ac$.
The solution set of a quadratic equation can include real numbers, imaginary numbers, or a combination of both, based on the discriminant's value.

coefficients

Coefficients are the numerical factors in the terms of a polynomial. In the quadratic equation $ax^2 + bx + c = 0$, the coefficients are *a*, *b*, and *c*.

*a* is the coefficient of $x^2$ and it determines the parabola's direction of opening.
*b* is the coefficient of $x$ and it impacts the parabola's position along the x-axis.
*c* is the constant term and it influences the y-intercept of the parabola.

Identifying these coefficients is crucial for calculating the discriminant and determining the nature of the equation's solutions.

imaginary solutions

Imaginary solutions occur when the discriminant $b^2 - 4ac$ is less than zero. This situation implies that the quadratic equation does not have real-number solutions, as the square root of a negative number is not a real number.
Instead, the solutions are in the form of complex numbers. Complex numbers consist of a real part and an imaginary part, usually expressed as $a + bi$, where *i* is the imaginary unit satisfying $i^2 = -1$. For example, the solutions for the equation $x^2 + 3x + 4 = 0$, with a discriminant of $-7$, would be complex.

real-number solutions

Real-number solutions occur under specific conditions of the discriminant:

**Two different real-number solutions**: When the discriminant $b^2 - 4ac$ is greater than zero, the quadratic equation has two distinct real-number solutions.
**One real-number solution**: When the discriminant is equal to zero, the quadratic equation has exactly one real-number solution, which is also known as a repeated or double root.

For example, if you have the equation $x^2 - 2x + 1 = 0$, the discriminant is $0$, resulting in one real-number solution $x = 1$. Such distinctions are vital in understanding the behavior and properties of quadratic equations.

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91影视

For each of the following, find the discriminant, \(b^{2}-4 a c\) and then determine whether one real-number solution, two different real-number solutions, or two different imaginary-number solutions exist. $$x^{2}+3 x+4=0$$

Short Answer

Step by step solution

- Identify coefficients

- Calculate the Discriminant

- Determine the Nature of the Solutions

Key Concepts

quadratic equation

coefficients

imaginary solutions

real-number solutions

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