Negative value of an even root?

[akselk]

I've got a question about the 2nd problem in chapter 5 of guide guide 3, page 79. The answer given is:

3d^2 + 18d + 27 + √(d+3)^2

= 3d^2 + 18d + 27 + d + 3 OR 3d^2 + 18d + 27 - (d + 3)

I thought that even roots only have positive values so how come there are two solutions for this?

[tpsharma2u]

3d^2 + 18d + 27 + √(d+3)^2...
hi u r right on ur pt but it will still have two values..
=> 3(d^2 + 6d + 9) + (d+3)= 3(d+3)^2+(d+3).....
=> (d+3)(3(d+3)+1))=(d+3)(3d+10)....
so d=-3 or -10/3

[georgepa]

It might help thinking of it this way:


x^2 = 4

Implies x = +/- √ 4 = +/- 2


Notice the +/- is before the √ sign indicating that the value from the √ is always positive.

Another example is in the quadratic formula to find the roots

Eqn: ax^2 + bx + c = 0

=> x = [-b ± √(b2 - 4ac)]/2a

Here also the +/- is before the √ which shows that the value of √(blah) is always +ve

[esledge]

When you take the square root of a plain-old number, not an expression, the GMAT always means the positive root. For example, sqrt(3^2) = sqrt(9) = 3, NOT -3. Similarly, sqrt[(-3)^2] = sqrt[9] = 3, NOT -3, even though -3 was the original base we squared inside the radical.

More generally, sqrt(x^2) = |x|.

In this problem, the question is what to do with sqrt[(d+3)^2]. Following the general rule above, sqrt[(d+3)^2] = |d+3|, which could be (d+3) or -(d+3), depending on the sign of (d+3).