/*! 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 45 Find a simplified form of \(f(x)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 45

Find a simplified form of \(f(x) .\) Assume that \(x\) can be any real number. $$f(x)=\sqrt{5 x^{2}-10 x+5}$$

Short Answer

Expert verified
\(f(x) = \sqrt{5} \times |x - 1|\)

Step by step solution

01

- Identify the Expression Under the Square Root

Notice that the expression under the square root is a quadratic expression: \[5x^2 - 10x + 5\]
02

- Factor the Quadratic Expression

Factor the quadratic expression \[5x^2 - 10x + 5\] by taking out the common factor of 5: \[= 5(x^2 - 2x + 1)\]
03

- Recognize a Perfect Square

Observe that the quadratic expression \[x^2 - 2x + 1\] is a perfect square trinomial.It can be rewritten as: \[= 5(x - 1)^2\]
04

- Apply the Square Root Property

Simplify the square root of the expression: \[f(x) = \sqrt{5(x - 1)^2}\]Use the property \[\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\]: \[f(x) = \sqrt{5} \times \sqrt{(x - 1)^2}\]
05

- Simplify Further

Since \[\sqrt{(x - 1)^2} = |x - 1|\] (the absolute value of \[x - 1\]):\[f(x) = \sqrt{5} \times |x - 1|\]

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 expression
A quadratic expression is a polynomial of degree 2, generally represented in the form: a^2 + bx + cIn our exercise, the quadratic expression is:5x^2 - 10x + 5. These expressions are bell-shaped curves when plotted and play a key role in algebra.
factoring
Factoring is the process of breaking down an expression into multiple factors that, when multiplied, give the original expression. In our case:5x^2 - 10x + 5We take out 5 as a common factor. This reduces the expression to:5(x^2 - 2x + 1).By factoring, we simplify the problem, making it easier to solve.
perfect square trinomial
A perfect square trinomial is a special type of quadratic expression with the form:(a-b)^2 = a^2 - 2ab + b^2.In our example, we identify x^2 - 2x + 1 as a perfect square trinomial, which can be rewritten as:(x-1)^2.Recognizing this form allows us to easily apply further properties.
square root property
The square root property states that \(\text{鈭殅(a \times b) = \text{鈭殅a \times \text{鈭殅b\)In our exercise, we have:\(\text{鈭殅(5(x-1)^2) = \text{鈭�5} \times \text{鈭�((x-1)^2)}\)This simplifies to: \(\text{鈭�5} \times |x-1|\).Using this property helps in simplifying complex expressions.
absolute value
Absolute value measures the distance a number is from zero, regardless of its direction. It is always positive. For our exercise, the absolute value of \( |x-1| \)ensures that the root expression is simplified positively. Thus, the final simplified form is:f(x) = \(\text{鈭�5} \times |x-1|\).

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 simplified form for \(f(x) .\) Assume \(x \geq 0\) $$ \begin{aligned} f(x)=& \sqrt{20 x^{2}+4 x^{3}}-3 x \sqrt{45+9 x} \\ &+\sqrt{5 x^{2}+x^{3}} \end{aligned} $$

The length and the width of a rectangle are given by consecutive integers. The area of the rectangle is \(90 \mathrm{cm}^{2} .\) Find the length of a diagonal of the rectangle.

Factor completely. $$w^{3}-4 w+3 w^{2}-12$$

f(x)\( and \)g(x)\( are as given. Find \)(f \cdot g)(x) \cdot$ Assume that all variables represent non-negative real numbers. $$ f(x)=x+\sqrt{7}, g(x)=x-\sqrt{7} $$

Is the set of real numbers a subset of the set of complex numbers? Why or why not?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

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.