My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Thanks Tree4Thanks
  • 1 Post By greg1313
  • 1 Post By v8archie
  • 2 Post By skipjack
Reply
 
LinkBack Thread Tools Display Modes
October 7th, 2018, 02:00 PM   #1
Senior Member
 
SenatorArmstrong's Avatar
 
Joined: Nov 2015
From: United States of America

Posts: 181
Thanks: 21

Math Focus: Calculus and Physics
Best way to factor this?

16x^2 + 8x

Looks like the solution is (4x+ 2)^2 - 1.

Do you arrive at this solution by completing the square? When I tried to complete square things started to go south. Any tips?

Thanks!
SenatorArmstrong is offline  
 
October 7th, 2018, 02:34 PM   #2
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,879
Thanks: 1087

Math Focus: Elementary mathematics and beyond
$$16x^2+8x=8x(2x+1)$$
Thanks from SenatorArmstrong

Last edited by greg1313; October 8th, 2018 at 12:55 AM.
greg1313 is offline  
October 7th, 2018, 03:02 PM   #3
Senior Member
 
SenatorArmstrong's Avatar
 
Joined: Nov 2015
From: United States of America

Posts: 181
Thanks: 21

Math Focus: Calculus and Physics
Quote:
Originally Posted by greg1313 View Post
$$16x^2+8x=8x(2x+1)$$


My apologies. I meant to type (4x+1)^2 - 1. Wolfram confirms that an an alternate solution. Is there a technique that can be used to arrive at this?

Last edited by greg1313; October 8th, 2018 at 12:56 AM.
SenatorArmstrong is offline  
October 7th, 2018, 03:19 PM   #4
Senior Member
 
SenatorArmstrong's Avatar
 
Joined: Nov 2015
From: United States of America

Posts: 181
Thanks: 21

Math Focus: Calculus and Physics
Seems like I need to "force" the expression into a perfect square trinomial. Which is done by adding 1 to both sides. Is there a "trick" to knowing you must add 1 to both sides to turn it into a perfect square trinomial? Or is this just something that comes with practice?

Thanks!
SenatorArmstrong is offline  
October 7th, 2018, 03:26 PM   #5
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,445
Thanks: 2499

Math Focus: Mainly analysis and algebra
\begin{align}16x^2+8x &= (4x)^2 + 2(4x) \\ &= u^2 + 2u &(u=4x) \\ &= \left(u+\tfrac22\right)^2 + c &(\text{constant $c$}) \\ &= \left(u + 1\right)^2 + c \\ &= \left(4x+1\right)^2 + c \\ \cancel{16x^2}+\cancel{8x} &= \cancel{16x^2} + \cancel{8x} + 1 + c \\ 0 &= 1 + c \\ c &= -1\\ \implies 16x^2+8x &= \left(4x+1\right)^2 - 1 \end{align}

This isn't a factorisation, it's the vertex form. A factorisation leaves a product of terms as in Greg's post.
Thanks from SenatorArmstrong

Last edited by v8archie; October 7th, 2018 at 04:12 PM.
v8archie is offline  
October 7th, 2018, 04:02 PM   #6
Senior Member
 
SenatorArmstrong's Avatar
 
Joined: Nov 2015
From: United States of America

Posts: 181
Thanks: 21

Math Focus: Calculus and Physics
Quote:
Originally Posted by v8archie View Post
\begin{align}16x^2+8x &= (4x)^2 + 2(4x) \\ &= u^2 + 2u &(u=4x) \\ &= \left(u+\tfrac22\right)^2 + c &(\text{constante $c$}) \\ &= \left(u + 1\right)^2 + c \\ &= \left(4x+1\right)^2 + c \\ \cancel{16x^2}+\cancel{8x} &= \cancel{16x^2} + \cancel{8x} + 1 + c \\ 0 &= 1 + c \\ c &= -1\\ \implies 16x^2+8x &= \left(4x+1\right)^2 - 1 \end{align}

This isn't a factorisation, it's the vertex form. A factorisation leaves a product of terms as in Greg's post.
Oh I see! Thanks a lot! Makes sense now! Have a good day!!
SenatorArmstrong is offline  
October 7th, 2018, 05:51 PM   #7
Global Moderator
 
Joined: Dec 2006

Posts: 19,713
Thanks: 1806

16x² + 8x = (4x+ 1)² - 1 = (4x + 1 - 1)(4x + 1 + 1) = 4x(4x + 2) = 8x(2x + 1)
skipjack is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
factor



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Factor ((L-m)/2)x² - ((L+m)/2) x + m =0 Mukunth Algebra 3 March 9th, 2016 12:08 AM
factor mared Algebra 5 April 23rd, 2014 04:11 AM
Factor Eminem_Recovery Algebra 11 June 19th, 2011 08:50 PM
how can these to be factor? haebin Calculus 2 September 14th, 2009 09:25 PM
Factor x^4 + 1 profetas Algebra 3 August 25th, 2009 09:20 AM





Copyright © 2018 My Math Forum. All rights reserved.