if 2x2 = 24 -x4 andwhich of the following can be the value of x?a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer?

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GMAT Exam > GMAT Questions > if 2x2 = 24 -x4 andwhich of the following can...

if 2x²= 24 -x⁴and

which of the following can be the value of x?

a)
I only
b)
II only
c)
III only
d)
I, II and III
e)
None of the above

Correct answer is option 'B'. Can you explain this answer?

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Verified Answer

if 2x2 = 24 -x4 andwhich of the following can be the value of x?a)I on...

Given:

Approach:

To answer the question, we need to know the possible values of x.
So, we’ll first solve Equation 1, then Equation 2 and then find the common roots of both these equations. These will be the possible values of x
Then, we’ll evaluate the 3 options.

Working Out:

Solving Equation I
- 2x²= 24 -x⁴
- We can factorize this equation as it is, but if powers like x⁴ intimidate you, then it’s better to substitute x² with a new variable, say z. That ways, the equation will look simpler: 2z = 24 – z²
- Rearranging the terms: z² + 2z – 24 = 0
- z² + 6z – 4z – 24 = 0
- z(z+6) – 4(z+6) = 0
- (z+6)(z-4) = 0
- This means, z = -6 or 4
- But z = x². So, x² = -6 or 4
  - Perfect square x² cannot be negative. Therefore, we can reject the negative root.
- So, x² = 4
- Therefore, x = +2 or -2
- Thus, Equation I has 2 roots: 2 and -2
Solving Equation II
- This equation looks difficult primarily because it involves a square root term, . So, for the ease of our calculations, let’s substitute =y
- So, the equation becomes:
- Multiplying both sides with 2y, we get: 2y² – 6 = y
- Rearranging the terms: 2y² – y – 6 = 0
- This means, y = -3/2 or y = 2
- Since y = we can write:
- However, since the square root of a number is always positive, the value of cannot be -3/2 .
- If you’ve any doubt about this, think: When you write z² = 16 and then take the square root on both sides, you write: Either z = √16 or z = -√16. Note here that, in the ‘or’ case, you put the minus sign outside the value of √16. This minus sign makes z negative (we get z = -4 in the ‘or’ case) but √16 is always positive (equal to 4).
- Similarly, if we have z² = x + 6, then on taking the square root on both sides, we will write: Either or
- z= - So, z may be positive or negative but will always be positive.
- So, =2
- Upon squaring both sides of this equation, we get: x + 6 = 4
- Therefore, x = -2
- Thus, we get a single value of x from Equation II
Finding the possible values of x
- From Equation I: x = 2 or -2
- From Equation II: x = -2
- So, the value of x that satisfies both equations: x = -2
- Thus, only 1 value of x is possible: {-2}
- Evaluating the 3 options
  - Out of the 3 given options, x can only be equal to -2
- Looking at the answer choices, we see that the correct answer is Option B

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