When is following equation true?

[Rectifier]

1. The problem

When is the following equation true
$\sqrt{c^2+14c+49} = c + 7$

a) for all real c
b) for $c \geq -7$
c) for $c < -7$
d) c > 0
e) c < 0

The attempt 1
I know that the root of $c^2+14c+49 = 0$ is $c = -7$ and that this sqr-root is only defined for positive numbers. Thus the equation is true only when the stuff below the root is positive. But that stuff is always positive...

The attempt 2
$\sqrt{c^2+14c+49} = c + 7 \\ \sqrt{(c+7)^2} = c + 7 \\ c+7 = c + 7 \\$
Thus this equation is true for all real c:s. But somehow this is wrong.

 

[andrewkirk]

Look at your last step in Attempt 2. Are you sure that it's always the case that $\sqrt{x^2}=x$, given that the convention is that the positive square root is always implied by the square root sign? What about if x=-1? What happens if you start with -1, square it and then take the square root (which is by convention positive). Do you end up with the number you started with?

 

[Rectifier]

What makes you you think your answer is wrong?

The answer in my book :)

 

[DrClaude]

$\sqrt{c^2+14c+49} = c + 7 \\ \sqrt{(c+7)^2} = c + 7 \\ c+7 = c + 7 \\$

The problem is in that last step, because $\sqrt{x^2} \neq x$. Can you see why?

The answer in my book :)

What about trying it yourself?

 

[andrewkirk]

Yeah, sorry, my first answer was too quick. I hope my redraft makes more sense.

 

[Rectifier]

The problem is in that last step, because $\sqrt{x^2} \neq x$. Can you see why?What about trying it yourself?

Yeah. Because negative x:es give different results.

How can I implement that in my problem?

 

[DrClaude]

Yeah. Because negative x:es give different results.

How can I implement that in my problem?

You have to use absolute values.

 

[Rectifier]

So basically |c + 7| = c + 7

 

[DrClaude]

So basically |c + 7| = c + 7

Yes. You should be able to convert that to a condition on $c$.

 

[Mark44]

So basically |c + 7| = c + 7

Yes. You should be able to convert that to a condition on $c$.

To be clear, |c + 7| is not equal to c + 7, as when, for example, c = -8. I believe that @DrClaude is in agreement with this, but the casual reader might misinterpret his comment.

$\sqrt{(c + 7)^2} \neq c + 7$
but
$\sqrt{(c + 7)^2} = |c + 7|$

 

[DrClaude]

To be clear, |c + 7| is not equal to c + 7, as when, for example, c = -8. I believe that @DrClaude is in agreement with this, but the casual reader might misinterpret his comment.

Of course I agree

My point is that the question starts with:

When is the following equation true
$\sqrt{c^2+14c+49} = c + 7$

which is simplified to:

When is the following equation true
$| c+ 7| = c + 7$

from which it is easy to get a condition on $c$ for the original equation to be true.

 

[HallsofIvy]

Use the definition of absolute value: $|a|= a$ if $a\ge 0$, $|a|= -a$ if $a< 0$.