/*! 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 96 Find \(a\) if \(f(a)=9\) and \(f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 96

Find \(a\) if \(f(a)=9\) and \(f(x)=2 x^{2}+1\)

Short Answer

Expert verified
a = 2 and a = -2

Step by step solution

01

- Understand the function

We are given the function \( f(x) = 2x^2 + 1 \) and need to find \( a \) such that \( f(a) = 9 \).
02

- Set up the equation

Since we need to find \( a \) such that \( f(a) = 9 \), we set up the equation: \( 2a^2 + 1 = 9 \).
03

- Solve for \( a \)

Solve the equation: \[ 2a^2 + 1 = 9 \] Subtract 1 from both sides: \[ 2a^2 = 8 \] Divide both sides by 2: \[ a^2 = 4 \] Take the square root of both sides: \[ a = \pm 2 \]
04

- Identify the solutions

Both \( a = 2 \) and \( a = -2 \) satisfy the equation \( 2a^2 + 1 = 9 \). So, the solutions are \( a = 2 \) and \( a = -2 \).

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.

headline of the respective core concept
Function evaluation is a fundamental concept in algebra. It involves substituting a given value into a function to find the corresponding output. For this exercise, we have the function \( f(x) = 2x^2 + 1 \). The problem asks us to find the value of \(a\) such that \( f(a) = 9 \). By plugging \(a\) back into the function, we establish an equation to solve for \(a\). This step is crucial as it translates the problem from a verbal statement into a manageable algebraic form.
headline of the respective core concept
Quadratic equation solving is essential when dealing with functions like \( f(x) = 2x^2 + 1 \). To find \(a\) for which \( f(a) = 9 \), we set up the equation: \(2a^2 + 1 = 9\). This is a quadratic equation in standard form \( Ax^2 + Bx + C = 0 \). Solving for \(a\) involves isolating \(a^2\) by first subtracting 1 from both sides: \( 2a^2 = 8 \). Then we divide by 2: \( a^2 = 4 \). Finally, taking the square root of both sides, we get two possible solutions: \(a = 2\) or \(a = -2\). Quadratic equations result in two solutions due to the square root property.
headline of the respective core concept
Algebraic manipulation is a key skill in simplifying and solving equations. In this exercise, once we set up the quadratic equation \(2a^2 + 1 = 9\), we use algebraic steps to isolate the variable \( a \). These steps include operations like addition, subtraction, multiplication, division, and taking square roots. Each step adheres to the properties of equality, ensuring the equation remains balanced. For instance, subtracting 1 from both sides simplifies the equation to \(2a^2 = 8\). Dividing both sides by 2 further simplifies it to \(a^2 = 4\). Taking the square root of both sides gives us the solutions \( a = \pm 2 \). These manipulations help us solve the equation accurately.

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

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ f(3) \cdot k(3) $$

Graph each of the following equations on a graphing calculator: Choose a viewing window that shows both the \(x\) - and \(y\) -intercepts, then draw the graph on paper with the \(x\) - and \(y\) -axes labeled appropriately. $$ y=999 x-100 $$

Graph each function by plotting points and state the domain and range. If you have a graphing calculator, use it to check your results. $$ y=5 $$

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ g(b) $$

Find the domain and range for the function \(f(x)=x^{2}+1\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

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.