For how many even integers x is x(x+1) + (x-18)(x+

Question

For how many even integers  x  is  x(x+1) + (x-18)(x+2) < (x-4)(x+1) ?

(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

suntprovident · Answer

I am stuck on how to proceed post getting the equation x2-12x-32<0 as there are no roots. If someone can help post this ?

Adit_ · Answer

It is quite simple after that, use the formula for quadratic roots:
-b+root(b^2-4ac)/2a, that formula which you usually use.

You will get 14.xx and -2.xx  .
Basically all even integers from -2 to 14.
You get 9 such numbers.

Why particularly this range? Because you will have negative values between the two roots as the parabola is facing upwards.
All values on either side of the roots will be positive.

So we have  9  values.

Hope it helps.

Edskore · Answer

Answer is (E) 9.

The key is to simplify the inequality algebraically before worrying about which integers work. A lot of people try plugging numbers here — it's slow and error-prone.

1. Expand the left side: x(x+1) + (x-18)(x+2) = x^2 + x + x^2 + 2x - 18x - 36 = 2x^2 - 15x - 36

2. Expand the right side: (x-4)(x+1) = x^2 + x - 4x - 4 = x^2 - 3x - 4

3. Bring everything to the left: (2x^2 - 15x - 36) - (x^2 - 3x - 4) < 0, which simplifies to x^2 - 12x - 32 < 0

4. Find the roots: discriminant = 144 + 128 = 272. sqrt(272) is between 16 and 17 (since 16^2 = 256 and 17^2 = 289), approximately 16.49. So roots are roughly (12 - 16.49)/2 ≈ -2.24 and (12 + 16.49)/2 ≈ 14.24.

5. Since the parabola opens upward, the inequality holds between the roots: -2.24 < x < 14.24.

6. Even integers strictly in this range: -2, 0, 2, 4, 6, 8, 10, 12, 14. Count = 9.

The classic trap is skipping -2 and starting the count from 0. Since -2 > -2.24, it's valid. I almost missed it the first time I worked through this type of problem.

Answer: E (9)

kevincan · Answer

Generally in the GMAT you would be able to factor that expression easily. Not in this case : however , you could use the quadratic formula or conclude that x2 - 12x - 32 = ( x - a)(x + b) where a is a bit greater than 14 and b a bit greater than 2. Thus x2 - 12x - 32 < 0 for all even numbers from -2 to 14.

For how many even integers x is x(x+1) + (x-18)(x+

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For how many even integers x is x(x+1) + (x-18)(x+

Prep Toolkit

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