Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 82

Chapter 4, Problem 82

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x²+4x−1=0

Verified step by step guidance

Identify the type of equation given. Here, the equation

x^{2} + 4x - 1 = 0

is a quadratic equation because it is a polynomial of degree 2.

Recall the quadratic formula, which is used to solve any quadratic equation of the form

ax^2 + bx + c = 0

. The formula is

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

Determine the coefficients from the equation:

a = 1

b = 4

, and

c = -1

Substitute the values of

a

b

, and

c

into the quadratic formula:

x = \(\frac{-4 \pm \sqrt{4^2 - 4(1)(-1)}\)}{2(1)}

Simplify the expression under the square root (the discriminant) and then simplify the entire expression to find the two possible values of

x

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the coefficients.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It uses the coefficients a, b, and c to find the roots, including complex solutions if the discriminant is negative.

Discriminant

The discriminant, given by b² - 4ac, determines the nature of the roots of a quadratic equation. If positive, there are two distinct real roots; if zero, one real root; and if negative, two complex conjugate roots.

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The Discriminant

Related Practice

Textbook Question

Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x⁴−3x²−4). If f(3)=−150, determine the value of a_n.

Textbook Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x²+1)/x

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Textbook Question

Solve the variation problems in Exercises 77–82. The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a wavelength of 2.4 feet?

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Textbook Question

Find the inverse of f(x)=(x−10)/(x+10).

Textbook Question

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x²+4x+6=0

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Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x−1=0

Verified video answer for a similar problem:

Key Concepts

Quadratic Equations

Quadratic Formula

Discriminant

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x²+4x−1=0