MGMAT Guide 3 Page 155 Explanation

[shoumik]

Hi,

I am kind of confused about the way a certain question was solved:

If (z+3)^2 = 25, what is z?

Shouldn't the conditions be:

+(z+3) = + 5
-(z+3) = + 5
+(z+3) = -5
-(z+3) = -5

This would give us results of z = {-2, -8, 2, 8}

But the book answer is only z = 2, -8.

Can someone explain the logic? I am following the logic explained in page 151 of MGMAT Guide 3, 4th edition.

[jnelson0612]

Hi,

I am kind of confused about the way a certain question was solved:

If (z+3)^2 = 25, what is z?

Shouldn't the conditions be:

+(z+3) = + 5
-(z+3) = + 5
+(z+3) = -5
-(z+3) = -5

This would give us results of z = {-2, -8, 2, 8}

But the book answer is only z = 2, -8.

Can someone explain the logic? I am following the logic explained in page 151 of MGMAT Guide 3, 4th edition.

Hi shoumik,
First, try testing the values of z you came up with in the original equation. You said possible values of z are -2, -8, 2, 8.

Here's the equation: (z+3)^2 = 25

If z=-2, (-2+3)^2 = 1, not 25. z cannot be -2.
If z=-8, (-8+3)^2 = 25. -8 works.
If z=2, (2+3)^2 = 25. 2 works.
If z=8, (8+3)^2 = 121, not 25. z cannot be 8.
Thus, only 2 and -8 are possible solutions.

However, I'm not sure how you got the -2 and 8 in the first place. Let's look at what you wrote:
+(z+3) = + 5 If I solve this, z would be 2
-(z+3) = + 5 z would be -8
+(z+3) = -5 z would be 2
-(z+3) = -5 z would be -8

So, nothing you've done conceptually is wrong; you just did not solve some of these equations properly.

Let's try to make this whole problem simpler by looking at the equation again:
(z+3)^2 = 25

or *something* squared = 25. What are the two possibilities? Either 5 squared is 25 or -5 squared is 25. Thus, I should set (z+3)=5 and solve for z, and (z+3)=-5 and solve for z. I get z=2 and z=-8.

I hope that this helps, and please let us know if we can explain further.

[shoumik]

Thanks for the reply. Regarding the solution of -2 and 8:

I think there might be an error in the way you calculated the last two equations:

1) + (z+3) = -5
= +z +3 = -5
= +z +3 -3 = -5 - 3
= z = -8 Not +2 as you stated.

2) - (z+3) = -5
-z-3 = -5
-z -3 +z +5 = -5 +5 +z
-3 + 5 = z
z = 2 Not -8 as you stated.

Please verify if I made a mistake in calculating my equations.

As for the solutions [2, -2, 8, -8], I should plug all the solutions into the equations to verify which answers do actually make right side = left side?

Another question: How would we do it in terms of inequality:

For example, x^2 > 9
|x| > |3| or |x| > 3 or x > |3|?

What would the solutions be in this case?

Thanks again.

[jlucero]

Good catch. She had those two solutions backwards, but it still gives you only two answers: 2 or -8. I would also point out with the four possibilities that you gave

+(z+3) = + 5
-(z+3) = + 5 -----> (z+3) = -5
+(z+3) = -5
-(z+3) = -5 -----> (z+3) = 5

The first and last are the same equation as are the second and third. One way to avoid this confusion in this case is to multiply both sides by negative one so you don't need to worry about moving the negative into the expression in parenthesis. This also shows that you can simplify these types of expressions by saying that (z+3) must be equal to a positive or negative five.

As for your second question- anytime I see an inequality with an even exponent or any absolute value question, I rewrite the expression twice and solve separately: once if the value is positive and once if the value is negative.

|x+5| > 3

1) If x is positive:

x+5 > 3
x > -2

*Which numbers are positive AND greater than -2? All positive numbers:

x > 0

2) If x is negative:

-(x+5) > 3
x+5 < -3
x < -8

*Which numbers are negative AND less than -8? All numbers less than -8:

x < -8

And these are your ranges. Solve for a positive and negative solution, then double check that your answers fit with your scenarios. You might also want to plug them back into the equation to make sure you did the math correct-- I've been doing this math for years and I still double check most of my work!