Question 149608
Technique #1  Factoring:


First let's factor {{{12x^2-10x-42}}}



{{{12x^2-10x-42}}} Start with the given expression



{{{2(6x^2-5x-21)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{6x^2-5x-21}}}


------------------------------------------------------------


Looking at the expression {{{6x^2-5x-21}}}, we can see that the first coefficient is {{{6}}}, the second coefficient is {{{-5}}}, and the last term is {{{-21}}}.



Now multiply the first coefficient {{{6}}} by the last term {{{-21}}} to get {{{(6)(-21)=-126}}}.



Now the question is: what two whole numbers multiply to {{{-126}}} (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 {{{-126}}} (the previous product).



Factors of {{{-126}}}:

1,2,3,6,7,9,14,18,21,42,63,126

-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126



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



These factors pair up and multiply to {{{-126}}}.

1*(-126)
2*(-63)
3*(-42)
6*(-21)
7*(-18)
9*(-14)
(-1)*(126)
(-2)*(63)
(-3)*(42)
(-6)*(21)
(-7)*(18)
(-9)*(14)


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>-126</font></td><td  align="center"><font color=black>1+(-126)=-125</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-63</font></td><td  align="center"><font color=black>2+(-63)=-61</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-42</font></td><td  align="center"><font color=black>3+(-42)=-39</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-21</font></td><td  align="center"><font color=black>6+(-21)=-15</font></td></tr><tr><td  align="center"><font color=black>7</font></td><td  align="center"><font color=black>-18</font></td><td  align="center"><font color=black>7+(-18)=-11</font></td></tr><tr><td  align="center"><font color=red>9</font></td><td  align="center"><font color=red>-14</font></td><td  align="center"><font color=red>9+(-14)=-5</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>126</font></td><td  align="center"><font color=black>-1+126=125</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>63</font></td><td  align="center"><font color=black>-2+63=61</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>42</font></td><td  align="center"><font color=black>-3+42=39</font></td></tr><tr><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>21</font></td><td  align="center"><font color=black>-6+21=15</font></td></tr><tr><td  align="center"><font color=black>-7</font></td><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>-7+18=11</font></td></tr><tr><td  align="center"><font color=black>-9</font></td><td  align="center"><font color=black>14</font></td><td  align="center"><font color=black>-9+14=5</font></td></tr></table>



From the table, we can see that the two numbers {{{9}}} and {{{-14}}} add to {{{-5}}} (the middle coefficient).



So the two numbers {{{9}}} and {{{-14}}} both multiply to {{{-126}}} <font size=4><b>and</b></font> add to {{{-5}}}



Now replace the middle term {{{-5x}}} with {{{9x-14x}}}. Remember, {{{9}}} and {{{-14}}} add to {{{-5}}}. So this shows us that {{{9x-14x=-5x}}}.



{{{6x^2+highlight(9x-14x)-21}}} Replace the second term {{{-5x}}} with {{{9x-14x}}}.



{{{(6x^2+9x)+(-14x-21)}}} Group the terms into two pairs.



{{{3x(2x+3)+(-14x-21)}}} Factor out the GCF {{{3x}}} from the first group.



{{{3x(2x+3)-7(2x+3)}}} Factor out {{{7}}} 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.



{{{(3x-7)(2x+3)}}} Combine like terms. Or factor out the common term {{{2x+3}}}



So {{{12x^2-10x-42}}} factors to {{{2(3x-7)(2x+3)}}}



{{{2(3x-7)(2x+3)=0}}} Set the factored expression equal to zero



Now set each factor equal to zero:


{{{3x-7=0}}} or  {{{2x+3=0}}} 


{{{x=7/3}}} or  {{{x=-3/2}}}    Now solve for x in each case



So our answers are 


 {{{x=7/3}}} or  {{{x=-3/2}}} 





<hr>



Technique #2 Quadratic Formula:




{{{12x^2-10x-42=0}}} Start with the given equation.



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-10) +- sqrt( (-10)^2-4(12)(-42) ))/(2(12))}}} Plug in  {{{a=12}}}, {{{b=-10}}}, and {{{c=-42}}}



{{{x = (10 +- sqrt( (-10)^2-4(12)(-42) ))/(2(12))}}} Negate {{{-10}}} to get {{{10}}}. 



{{{x = (10 +- sqrt( 100-4(12)(-42) ))/(2(12))}}} Square {{{-10}}} to get {{{100}}}. 



{{{x = (10 +- sqrt( 100--2016 ))/(2(12))}}} Multiply {{{4(12)(-42)}}} to get {{{-2016}}}



{{{x = (10 +- sqrt( 100+2016 ))/(2(12))}}} Rewrite {{{sqrt(100--2016)}}} as {{{sqrt(100+2016)}}}



{{{x = (10 +- sqrt( 2116 ))/(2(12))}}} Add {{{100}}} to {{{2016}}} to get {{{2116}}}



{{{x = (10 +- sqrt( 2116 ))/(24)}}} Multiply {{{2}}} and {{{12}}} to get {{{24}}}. 



{{{x = (10 +- 46)/(24)}}} Take the square root of {{{2116}}} to get {{{46}}}. 



{{{x = (10 + 46)/(24)}}} or {{{x = (10 - 46)/(24)}}} Break up the expression. 



{{{x = (56)/(24)}}} or {{{x =  (-36)/(24)}}} Combine like terms. 



{{{x = 7/3}}} or {{{x = -3/2}}} Simplify. 



So our answers are {{{x = 7/3}}} or {{{x = -3/2}}} 

  


<hr>


Technique # 3 Completing the square



{{{12 x^2+10 x-42}}} Start with the given expression



{{{12(x^2+(5/6)x-7/2)}}} Factor out the leading coefficient {{{12}}}



Take half of the x coefficient {{{5/6}}} to get {{{5/12}}} (ie {{{(1/2)(5/6)=5/12}}}).


Now square {{{5/12}}} to get {{{25/144}}} (ie {{{(5/12)^2=(5/12)(5/12)=25/144}}})





{{{12(x^2+(5/6)x+25/144-25/144-7/2)}}} Now add and subtract this value inside the parenthesis. Notice how {{{25/144-25/144=0}}}. Since we're adding 0, we're not changing the equation.




{{{12((x+5/12)^2-25/144-7/2)}}} Now factor {{{x^2+(5/6)x+25/144}}} to get {{{(x+5/12)^2}}}



{{{12((x+5/12)^2-529/144)}}} Combine like terms



{{{12(x+5/12)^2+12(-529/144)}}} Distribute



{{{12(x+5/12)^2-529/12}}} Multiply




So after completing the square, {{{12x^2+10x-42}}} becomes {{{12(x+5/12)^2-529/12}}}.



So {{{12x^2+10x-42=0}}} is equivalent to {{{12(x+5/12)^2-529/12=0}}}



{{{12(x+5/12)^2-529/12=0}}} Start with completed square equation.




{{{12(x+5/12)^2=529/12}}} Add {{{529/12}}} to both sides.



{{{(x+5/12)^2=529/144}}} Divide both sides by 12.



{{{x+5/12=0+-sqrt(529/144)}}} Take the square root of both sides.



{{{x+5/12=sqrt(529/144)}}} or {{{x+5/12=-sqrt(529/144)}}} Break up the expression



{{{x+5/12=23/12}}} or {{{x+5/12=-23/12}}} Take the square root of {{{529/144}}} to get {{{23/12}}}




{{{x=23/12-5/12}}} or {{{x=-23/12-5/12}}} Subtract {{{5/12}}} from both sides.



{{{x=3/2}}} or {{{x=-7/3}}} Combine like terms and simplify.



So the answers are {{{x=3/2}}} or {{{x=-7/3}}} 




<hr>



Technique # 4  Graphing


Simply graph {{{y=12 x^2+10 x-42}}} to get 



{{{ graph(500,500,-10,10,-10,10,0, 12x^2-10x-42) }}} Graph of {{{y=12 x^2+10 x-42}}}



Now use the calculator's zero function to find the zeros at {{{x=2.333}}} and {{{x=-1.5}}}