Question 151702
{{{45x^2-60x+20}}} Start with the given expression



{{{5(9x^2-12x+4)}}} Factor out the GCF {{{5}}}



Now let's focus on the inner expression {{{9x^2-12x+4}}}



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Looking at the expression {{{9x^2-12x+4}}}, we can see that the first coefficient is {{{9}}}, the second coefficient is {{{-12}}}, and the last term is {{{4}}}.



Now multiply the first coefficient {{{9}}} by the last term {{{4}}} to get {{{(9)(4)=36}}}.



Now the question is: what two whole numbers multiply to {{{36}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-12}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{36}}} (the previous product).



Factors of {{{36}}}:

1,2,3,4,6,9,12,18,36

-1,-2,-3,-4,-6,-9,-12,-18,-36



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



These factors pair up and multiply to {{{36}}}.

1*36
2*18
3*12
4*9
6*6
(-1)*(-36)
(-2)*(-18)
(-3)*(-12)
(-4)*(-9)
(-6)*(-6)


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-12}}}:



<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>36</font></td><td  align="center"><font color=black>1+36=37</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>2+18=20</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>3+12=15</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>4+9=13</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>6+6=12</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-36</font></td><td  align="center"><font color=black>-1+(-36)=-37</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-18</font></td><td  align="center"><font color=black>-2+(-18)=-20</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>-3+(-12)=-15</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-9</font></td><td  align="center"><font color=black>-4+(-9)=-13</font></td></tr><tr><td  align="center"><font color=red>-6</font></td><td  align="center"><font color=red>-6</font></td><td  align="center"><font color=red>-6+(-6)=-12</font></td></tr></table>



From the table, we can see that the two numbers {{{-6}}} and {{{-6}}} add to {{{-12}}} (the middle coefficient).



So the two numbers {{{-6}}} and {{{-6}}} both multiply to {{{36}}} <font size=4><b>and</b></font> add to {{{-12}}}



Now replace the middle term {{{-12x}}} with {{{-6x-6x}}}. Remember, {{{-6}}} and {{{-6}}} add to {{{-12}}}. So this shows us that {{{-6x-6x=-12x}}}.



{{{9x^2+highlight(-6x-6x)+4}}} Replace the second term {{{-12x}}} with {{{-6x-6x}}}.



{{{(9x^2-6x)+(-6x+4)}}} Group the terms into two pairs.



{{{3x(3x-2)+(-6x+4)}}} Factor out the GCF {{{3x}}} from the first group.



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



{{{(3x-2)^2}}} Condense the common factors.



So the expression goes from {{{5(9x^2-12x+4)}}} and factors further to {{{5(3x-2)^2}}}





So {{{45x^2-60x+20}}} factors to {{{5(3x-2)^2}}}.




{{{5(3x-2)^2=0}}} Set the factored expression equal to zero



{{{(3x-2)^2=0}}} Divide both sides by 5.



{{{3x-2=0}}} Take the square root of both sides



{{{3x=0+2}}} Add {{{2}}} to both sides.



{{{3x=2}}} Combine like terms on the right side.



{{{x=(2)/(3)}}} Divide both sides by {{{3}}} to isolate {{{x}}}.



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Answer:


So the answer is {{{x=2/3}}} 



Which approximates to {{{x=0.667}}}