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if (x+a)^2=x^2+10x+25 what is the value of a?

Guest Aug 6, 2017
 #1
avatar+7348 
+1

if (x+a)^2=x^2+10x+25 what is the value of a?

 

\((x+a)^2=x^2+10x+25\\ (x+{\color{blue}a})^2=(x+\color{blue}5)^2\\ \color{blue}a=5\)

 

laugh  !

asinus  Aug 6, 2017
 #2
avatar+2117 
+2

I would argue that there is a second solution to this problem, however. Let me demonstrate why:
 

\((x+a)^2=x^2+10x+25\) Take the square root of both sides of the equation. Of course, taking the square root of a number results in a positive and a negative answer.
\(x+a=\pm\sqrt{x^2+10x+25}\) Let's split these solutions into 2
\(x+a=\sqrt{x^2+10x+25}\) \(x+a=-\sqrt{x^2+10x+25}\)

 

 
   

 

Now, let's solve each equation separately:

 

\(x+a=\sqrt{x^2+10x+25}\) The first step is to factor the expression x^2+10x+25. Now, this trinomial is indeed a perfect-square trinomial. I know this because x^2 and 25 are perfect squares; x and 5. If you multiply the sum of x and 5 by 2, then you get 10x. If this condition is ever true with a trinomial, you have a perfect-square trinomial.
\(x+a=\sqrt{(x+5)^2}\) The square root and the square cancel each other out.
\(x+a=x+5\) Subtract x on both sides.
\(a=5\)  
   

 

You will notice that the user above also got this answer of a=5. What about the second solution? Well, you'll see. Solve for a in the following equation \(x+a=-\sqrt{x^2+10x+25}\), the second case:

 

\(x+a=-\sqrt{x^2+10x+25}\) We have already determined previously that \(\sqrt{x^2+10x+25}\) equals \(x+5\). Let's just plug that in to save a few steps.
\(x+a=-(x+5)\) Distribute the negative sign to both terms.
\(x+a=-x-5\) Subtract x on both sides.
\(a=-2x-5\)  
   

 

Is this actually a solution, though? If you are ever unsure of whether or not a solution is truly a solution, plug it into the original equation:

 

\((x+a)^2=x^2+10x+25\) Substitute a for -2x-5
\((x+(-2x-5))^2=x^2+10x+25\) Combine the terms of x and -2x
\((-x-5)^2=x^2+10x+25\) Expand (-x-5)^2 using the rule that \((a+b)^2=a^2+2ab+b^2\)
\(x^2+10x+25=x^2+10x+25\) Both sides of the equation are the same, so it should be clear now that both are equivalent. Therefore, -2x-5 is a solution.
   

 

Therefore, there are two solutions

 

\(a_1=5\)

\(a_2=-2x-5\)

TheXSquaredFactor  Aug 6, 2017

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