How many solutions does the equation 0.25x4 – 0.75x2 -1 = 0 have?

Question

How many solutions does the equation  0.25x^4  –  0.75x^2  -1 = 0 have?

A) 0
B) 1
C) 2
D) 3
E) 4

sujoykrdatta · Accepted Answer

Multiply throughout by 4:  x^4  –  3x^2  - 4 = 0

Assume  x^2 = k :  k^2 - 3k - 4 = 0

=>  (k - 4)(k + 1) = 0

=> k = 4 or -1

=>  x^2 = 4  (since  x^2  cannot be -1 for any real x)

=> x = 2 or -2

Answer C

exc4libur · Answer

x^4/4-3x^2/4-1=0
x^4-3x^2-4=0
x^2x^2-3x^2-4=0
(x^2)^2-3(x^2)-4=0
a^2-3a-4=0
(a-4)(a+1)=0
a=4 or a=-1
x^2=-1, cannot square root negative
x^2=4, |x|=2 or -2

(C)

planetearth9 · Answer

The question does not specify if x is real. In that case, would it be wrong to assume that x could be imaginary as well?

Bunuel · Answer

All numbers on the GMAT are real numbers by default.

dhichawla · Answer

There are 2 tricks with this question:
- First is to convert the co-efficients in the equation to whole numbers. Converts the equation to x^4 - 3x^2 - 4 = 0
- Second to substitute x^2 as another variable. This turns it into a quadratic equation which is much simpler to solve => (x^2 = a) => a^2 - 3a -4 = 0

We would be spending much more time to get to the actual answer without applying the above tricks.

omavsp · Answer

can you guys please help me understand how you got the -4?

dhichawla · Answer

Hi  omavsp ,

If you look at the original equation, it is 0.25x4 – 0.75x2 -1 = 0.
There are 2 ways to go about it:
1) Multiply the entire equation by 4. This being done to change the co-efficient of x^4 from 0.25 to 1.
 So, 4 * (0.25x4 – 0.75x2 -1 = 0) ==> 4*0.25x4 - 4*0.75x2 - 4*1 = 0 ==> x4 - 3x2 - 4 = 0.

2) If you convert the decimals to fractions in the original equation, you will end up with below:
 0.25x4 – 0.75x2 -1 = 0 ==> 25/100* x4 - 75/100* x2 - 1 = 0
==> x4/4 - 3x2/4 - 1 = 0 
Now, multiply by 4 again, you get ==> x4 - 3x2 - 4 = 0.

Hope this clarifies you doubt.

jcerdae · Answer

Hi!

I would like to know if my procedure is right! could someone check it (please)

1)Used sqrt(b^2 - 4ac) (also called discriminant) to identify how many solutions for X are
a) Discriminant >0; then there are 2 different solutions
b) Discriminant =0; then there are 2 equal solutions
c) Discriminant <0; then there are 0 solutions in the Real numbers world

The solution gave me a) meaning that there are 2 solutions

Kinshook · Answer

Asked: How many solutions does the equation  0.25x^4  –  0.75x^2  -1 = 0 have?

Let us multiple the whole equation by 4

xˆ4 - 3xˆ2 - 4 = 0
Let t = xˆ2

tˆ2 - 3t - 4= 0
(t-4)(t+1) = 0

Case 1: t = xˆ2 = 4
x = {-2,2}: 2 real solutions

Case 2: t= xˆ2 = - 1
x = {i, - i} : 2 imaginary solutions but No real solutions

Combining x = {-2,2 }: 2 real solutions

IMO C

Kushchokhani · Answer

jcerdae

The right term is not "discriminant" but "determinant". Further, we don't require to take sqrt of  b^2 - 4ac  because determinant is itself  b^2 - 4ac .

IMO, we can use this approach for the ques in hand.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

How many solutions does the equation 0.25x4 – 0.75x2 -1 = 0 have?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

How many solutions does the equation 0.25x4 – 0.75x2 -1 = 0 have?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards