/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 75 Find the zeros of the function a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 75

Find the zeros of the function algebraically. Give exact answers. $$f(x)=x^{2}-5 x+1$$

Short Answer

Expert verified
The zeros are \( x = \frac{5 + \sqrt{21}}{2} \) and \( x = \frac{5 - \sqrt{21}}{2} \).

Step by step solution

01

- Identify the Quadratic Formula

To find the zeros of a quadratic function, use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this formula, \( a \), \( b \), and \( c \) are coefficients from the standard quadratic form \( ax^2 + bx + c \).
02

- Determine Coefficients

For the function \( f(x) = x^2 - 5x + 1 \), identify the coefficients:\[ a = 1, \ b = -5, \ c = 1 \]
03

- Substitute Coefficients into Quadratic Formula

Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula:\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
04

- Simplify the Expression Inside the Square Root

Simplify the expression inside the square root (the discriminant):\[ x = \frac{5 \pm \sqrt{25 - 4}}{2} \]\[ x = \frac{5 \pm \sqrt{21}}{2} \]
05

- Simplify the Entire Expression

Divide the terms in the numerator by the denominator to get the final zeros:\[ x = \frac{5 + \sqrt{21}}{2} \] \[ x = \frac{5 - \sqrt{21}}{2} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

quadratic formula
The quadratic formula is a powerful tool used to find the zeros (or roots) of a quadratic equation. A quadratic equation is typically written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula works for all quadratic equations and guarantees solutions if they exist. To use this formula, simply identify the coefficients \( a \), \( b \), and \( c \) from your quadratic equation, substitute them into the formula, and then simplify. It covers all cases: real, repeated, and complex roots.
discriminant
The discriminant is a part of the quadratic formula located under the square root: \( \sqrt{b^2 - 4ac} \). It is crucial because it determines the nature of the roots of the quadratic equation. There are three primary cases based on the value of the discriminant:
  • If the discriminant \( b^2 - 4ac \) is greater than 0, there are two distinct real roots.
  • If the discriminant is equal to 0, there is exactly one real root (also called a repeated or double root).
  • If the discriminant is less than 0, the equation has two complex roots.
In our example with the quadratic function \( f(x) = x^2 - 5x + 1 \), the discriminant is calculated as follows: \[ b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot 1 = 25 - 4 = 21 \] Since 21 is greater than 0, this quadratic equation has two distinct real roots.
solving quadratic equations
To solve a quadratic equation and find its zeros, follow these steps:
  • Write in Standard Form: Ensure the quadratic equation is in the form \( ax^2 + bx + c = 0 \).
  • Identify Coefficients: Recognize the values of \( a \), \( b \), and \( c \).
  • Apply the Quadratic Formula: Substitute these coefficients into the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • Simplify the Solution: Calculate the discriminant and simplify the expression to find the zeros.
For the function \( f(x) = x^2 - 5x + 1 \), after applying the quadratic formula, we get: \[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \] The next step is to simplify this: \[ x = \frac{5 \pm \sqrt{25 - 4}}{2} = \frac{5 \pm \sqrt{21}}{2} \] And, finally, we get the two roots: \[ x = \frac{5 + \sqrt{21}}{2} \ \text{and} \ x = \frac{5 - \sqrt{21}}{2} \] These are the zeros of the quadratic function, meaning they are the values of \( x \) that make \( f(x) = 0 \).

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find a quadratic function with vertex \((4,-5)\) and containing the point \((-3,1)\)

a) Find the vertex. b) Determine whether there is a maximum or a minimum value and find that value. c) Find the range. d) Find the intervals on which the function is increasing and the intervals on which the function is decreasing. $$f(x)=-3 x^{2}+24 x-49$$

Determine whether the statement is true or false. The conjugate of a sum is the sum of the conjugates of the individual complex numbers.

Fill in the blank with the correct term. Some of the given choices will not be used. distance formula, midpoint formula, function, relation, \(x\) -intercept, y-intercept, perpendicular, parallel ,horizontal lines, vertical lines,symmetric with respect to the \(x\) -axis, symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, increasing, decreasing, constant For an equation \(y=f(x),\) if replacing \(x\) with \(-x\) produces an equivalent equation, then the graph is _____.

Solve and write interval notation for the solution set. Then graph the solution set. $$|7-x| \geq-4$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.