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 83

Chapter 4, Problem 83

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

Verified step by step guidance

Identify the type of equation given. Here, the equation is a quadratic equation of the form

x 2 + 4x + 6 = 0

, where the highest power of

x

is 2.

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

ax 2 + bx + c = 0

. The formula is

= rac{-b \(\u\)00b1 21212121b^2 - 4ac}{2a}

Identify the coefficients from the equation:

a = 1

b = 4

, and

c = 6

Substitute the values of

a

b

, and

c

into the quadratic formula:

x = rac{-4 21212121 21212121 4^2 - 4(1)(6)}{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 polynomial equation of degree two, generally written as ax² + bx + c = 0. It represents a parabola when graphed, and solving it means finding the values of x that satisfy the equation.

Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form (x + p)² = q. This technique transforms the quadratic into a perfect square trinomial, making it easier to solve for x.

Quadratic Formula

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

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Related Practice

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+x−6)/(x−3)

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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.