Question 285345
You made an error in thinking that {{{x*x=2x}}}. This is NOT true, and is a big misconception. So {{{x*x<>2x}}} for all 'x'. What you were thinking of (I think) is that {{{x+x=2x}}} which is true for all 'x'.

 


There are two ways to solve this problem.


# 1 Factoring


First factor {{{x^2 + 6x + 8}}}





Looking at the expression {{{x^2+6x+8}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{6}}}, and the last term is {{{8}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{8}}} to get {{{(1)(8)=8}}}.



Now the question is: what two whole numbers multiply to {{{8}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{6}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{8}}} (the previous product).



Factors of {{{8}}}:

1,2,4,8

-1,-2,-4,-8



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



These factors pair up and multiply to {{{8}}}.

1*8 = 8
2*4 = 8
(-1)*(-8) = 8
(-2)*(-4) = 8


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{6}}}:



<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>8</font></td><td  align="center"><font color=black>1+8=9</font></td></tr><tr><td  align="center"><font color=red>2</font></td><td  align="center"><font color=red>4</font></td><td  align="center"><font color=red>2+4=6</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>-1+(-8)=-9</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-2+(-4)=-6</font></td></tr></table>



From the table, we can see that the two numbers {{{2}}} and {{{4}}} add to {{{6}}} (the middle coefficient).



So the two numbers {{{2}}} and {{{4}}} both multiply to {{{8}}} <font size=4><b>and</b></font> add to {{{6}}}



Now replace the middle term {{{6x}}} with {{{2x+4x}}}. Remember, {{{2}}} and {{{4}}} add to {{{6}}}. So this shows us that {{{2x+4x=6x}}}.



{{{x^2+highlight(2x+4x)+8}}} Replace the second term {{{6x}}} with {{{2x+4x}}}.



{{{(x^2+2x)+(4x+8)}}} Group the terms into two pairs.



{{{x(x+2)+(4x+8)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+2)+4(x+2)}}} 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.



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





So {{{x^2+6x+8}}} factors to {{{(x+4)(x+2)}}}.



Since {{{x^2+6x+8=(x+4)(x+2)}}} and {{{x^2+6x+8=0}}}, this means that {{{(x+4)(x+2)=0}}}





So we've gone from {{{x^2+6x+8=0}}} to {{{(x+4)(x+2)=0}}}




Now remember that if AB = 0 (A times B equals zero), then either A=0, B=0 or both are equal to zero. This is the zero product property.



So use the zero product property to break {{{(x+4)(x+2)=0}}} down into the following equations shown below:


{{{x+4=0}}} or {{{x+2=0}}}



Now solve each equation to get 


{{{x=-4}}} or {{{x=-2}}}



So the two solutions are {{{x=-4}}} or {{{x=-2}}}



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Method # 2: Quadratic Formula





{{{x^2+6x+8=0}}} Start with the given equation.



Notice that the quadratic {{{x^2+6x+8}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=1}}}, {{{B=6}}}, and {{{C=8}}}



Let's use the quadratic formula to solve for "x":



{{{x = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{x = (-(6) +- sqrt( (6)^2-4(1)(8) ))/(2(1))}}} Plug in  {{{A=1}}}, {{{B=6}}}, and {{{C=8}}}



{{{x = (-6 +- sqrt( 36-4(1)(8) ))/(2(1))}}} Square {{{6}}} to get {{{36}}}. 



{{{x = (-6 +- sqrt( 36-32 ))/(2(1))}}} Multiply {{{4(1)(8)}}} to get {{{32}}}



{{{x = (-6 +- sqrt( 4 ))/(2(1))}}} Subtract {{{32}}} from {{{36}}} to get {{{4}}}



{{{x = (-6 +- sqrt( 4 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (-6 +- 2)/(2)}}} Take the square root of {{{4}}} to get {{{2}}}. 



{{{x = (-6 + 2)/(2)}}} or {{{x = (-6 - 2)/(2)}}} Break up the expression. 



{{{x = (-4)/(2)}}} or {{{x =  (-8)/(2)}}} Combine like terms. 



{{{x = -2}}} or {{{x = -4}}} Simplify. 



So the solutions are {{{x = -2}}} or {{{x = -4}}} 

  
  
Note: the order of the solutions does not matter.