Question 472964
Completing the square:


{{{2m^2 + 5m = 12}}} Start with the given equation.



{{{2m^2 + 5m - 12=0}}} Subtract 12 from both sides.




Now let's complete the square for the left side.



{{{2m^2+5m-12}}} Start with the given expression.



{{{2(m^2+(5/2)m-6)}}} Factor out the {{{m^2}}} coefficient {{{2}}}. This step is very important: the {{{m^2}}} coefficient <font size=4><b>must</b></font> be equal to 1.



Take half of the {{{m}}} coefficient {{{5/2}}} to get {{{5/4}}}. In other words, {{{(1/2)(5/2)=5/4}}}.



Now square {{{5/4}}} to get {{{25/16}}}. In other words, {{{(5/4)^2=(5/4)(5/4)=25/16}}}



{{{2(m^2+(5/2)m+highlight(25/16-25/16)-6)}}} Now add <font size=4><b>and</b></font> subtract {{{25/16}}} inside the parenthesis. Make sure to place this after the "m" term. Notice how {{{25/16-25/16=0}}}. So the expression is not changed.



{{{2((m^2+(5/2)m+25/16)-25/16-6)}}} Group the first three terms.



{{{2((m+5/4)^2-25/16-6)}}} Factor {{{m^2+(5/2)m+25/16}}} to get {{{(m+5/4)^2}}}.



{{{2((m+5/4)^2-121/16)}}} Combine like terms.



{{{2(m+5/4)^2+2(-121/16)}}} Distribute.



{{{2(m+5/4)^2-121/8}}} Multiply.



So after completing the square, {{{2m^2+5m-12}}} transforms to {{{2(m+5/4)^2-121/8}}}. So {{{2m^2+5m-12=2(m+5/4)^2-121/8}}}.



So {{{2m^2+5m-12=0}}} is equivalent to {{{2(m+5/4)^2-121/8=0}}}.



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Now let's solve {{{2(m+5/4)^2-121/8=0}}}



{{{2(m+5/4)^2-121/8=0}}} Start with the given equation.



{{{2(m+5/4)^2=0+121/8}}} Add {{{121/8}}} to both sides.



{{{2(m+5/4)^2=121/8}}} Combine like terms.



{{{(m+5/4)^2=(121/8)/(2)}}} Divide both sides by {{{2}}}.



{{{(m+5/4)^2=121/16}}} Reduce.



{{{m+5/4=""+-sqrt(121/16)}}} Take the square root of both sides.



{{{m+5/4=sqrt(121/16)}}} or {{{m+5/4=-sqrt(121/16)}}} Break up the "plus/minus" to form two equations.



{{{m+5/4=11/4}}} or {{{m+5/4=-11/4}}}  Take the square root of {{{121/16}}} to get {{{11/4}}}.



{{{m=-5/4+11/4}}} or {{{m=-5/4-11/4}}} Subtract {{{5/4}}} from both sides.



{{{m=3/2}}} or {{{m=-4}}} Combine like terms.



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Answer:



So the solutions are {{{m=3/2}}} or {{{m=-4}}}.




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Factoring:


{{{2m^2 + 5m = 12}}} Start with the given equation.



{{{2m^2 + 5m - 12=0}}} Subtract 12 from both sides.



Now let's factor:






Looking at the expression {{{2m^2+5m-12}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{5}}}, and the last term is {{{-12}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{-12}}} to get {{{(2)(-12)=-24}}}.



Now the question is: what two whole numbers multiply to {{{-24}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{5}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-24}}} (the previous product).



Factors of {{{-24}}}:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-24}}}.

1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{5}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-24</font></td><td  align="center"><font color=black>1+(-24)=-23</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>2+(-12)=-10</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>3+(-8)=-5</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>4+(-6)=-2</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>24</font></td><td  align="center"><font color=black>-1+24=23</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-2+12=10</font></td></tr><tr><td  align="center"><font color=red>-3</font></td><td  align="center"><font color=red>8</font></td><td  align="center"><font color=red>-3+8=5</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-4+6=2</font></td></tr></table>



From the table, we can see that the two numbers {{{-3}}} and {{{8}}} add to {{{5}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{8}}} both multiply to {{{-24}}} <font size=4><b>and</b></font> add to {{{5}}}



Now replace the middle term {{{5m}}} with {{{-3m+8m}}}. Remember, {{{-3}}} and {{{8}}} add to {{{5}}}. So this shows us that {{{-3m+8m=5m}}}.



{{{2m^2+highlight(-3m+8m)-12}}} Replace the second term {{{5m}}} with {{{-3m+8m}}}.



{{{(2m^2-3m)+(8m-12)}}} Group the terms into two pairs.



{{{m(2m-3)+(8m-12)}}} Factor out the GCF {{{m}}} from the first group.



{{{m(2m-3)+4(2m-3)}}} Factor out {{{4}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(m+4)(2m-3)}}} Combine like terms. Or factor out the common term {{{2m-3}}}



So {{{2m^2+5m-12}}} factors to {{{(m+4)(2m-3)}}}.



In other words, {{{2m^2+5m-12=(m+4)(2m-3)}}}.




So {{{2m^2+5m-12=0}}} turns into {{{(m+4)(2m-3)=0}}}



{{{(m+4)(2m-3)=0}}} Start with the given equation



{{{m+4=0}}} or {{{2m-3=0}}} Use the zero product property



{{{m=-4}}} or {{{m=3/2}}} Solve for 'm' in each equation.



So the solutions are {{{m=-4}}} or {{{m=3/2}}} (which are the same as the ones above)