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October 7th, 2018, 02:00 PM  #1 
Senior Member Joined: Nov 2015 From: United States of America Posts: 198 Thanks: 25 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! 
October 7th, 2018, 02:34 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond 
$$16x^2+8x=8x(2x+1)$$
Last edited by greg1313; October 8th, 2018 at 12:55 AM. 
October 7th, 2018, 03:02 PM  #3 
Senior Member Joined: Nov 2015 From: United States of America Posts: 198 Thanks: 25 Math Focus: Calculus and Physics  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. 
October 7th, 2018, 03:19 PM  #4 
Senior Member Joined: Nov 2015 From: United States of America Posts: 198 Thanks: 25 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! 
October 7th, 2018, 03:26 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 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. Last edited by v8archie; October 7th, 2018 at 04:12 PM. 
October 7th, 2018, 04:02 PM  #6  
Senior Member Joined: Nov 2015 From: United States of America Posts: 198 Thanks: 25 Math Focus: Calculus and Physics  Quote:
 
October 7th, 2018, 05:51 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,967 Thanks: 2216 
16x² + 8x = (4x+ 1)²  1 = (4x + 1  1)(4x + 1 + 1) = 4x(4x + 2) = 8x(2x + 1)


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