In the equation x^2 − cx + c = 0, c is a constant, and x is a variable

Question

In the equation x^2 − cx + c = 0, c is a constant, and x is a variable. How many different roots does x^2 − cx + c = 0 have?

(1) c < 0

(2) c = −1

ArunSharma12 · Answer

x^2 - cx + c = 0 
 x = \frac{c ± \sqrt{c^2 - 4c}}{ 2} 
maximum possible values for x are two.

statement 1: c < 0
 c^2 - 4c = c^2 + 4c > 0 ; implies roots are real and distinct. we'll have two roots.

statement 2: c = -1
 c^2 - 4c = (-1)^2 - (4 	imes -1) = 1 + 4 > 0 ; implies roots are real and distinct. we'll have two roots.

Both are sufficient

yashikaaggarwal · Answer

x^2 − cx + c = 0

Statement 1: c<0
since C is negative,
 x^2 + cx - c = 0
discriminant = b^2-4ac
=> C^2+4C > 0
(Hence roots will be two distinct positive roots)
(Sufficient)

Statement 2: c = -1
since C is negative,
 x^2 + x - 1 = 0
discriminant = b^2-4ac
=> 1^2+4 > 0
(Hence roots will be two distinct positive roots)
(Sufficient)

Answer is D

HoneyLemon · Answer

x^2 − cx + c = 0 Statement 1: c<0 D = b^2-4ac => C^2+4C > 0 Hence roots will be two distinct positive roots St 1 is Sufficient Statement 2: c = -1 Even C <1 so St 2 is included in statement 1 . So with C = -1, roots will be two distinct positive roots St 2 is Sufficient so my ans is D

andyjump · Answer

in statement 1 , please explain why we are considering c^2+4c > 0?

Thanks in adavance

HoneyLemon · Answer

andyjump

Consider below quadratic equation

ax^2 + bx + c = 0

In what scenario the quadratic will have real solution . 
 The ans is, if the discriminant D of the equation is greater than zero .

ax^2 + bx + c = 0

discriminant D >0 =>

b^2 - 4ac > 0  where b , a , c are constants .

in statement 1 .. in this equation 
 x^2 − cx + c = 0

c<0 (given)

So 
D =  b^2-4ac  =>  (-C)^2- 4(-C)  > 0

=>  C^2+4C  > 0

Hope it helps..

Dilpreetgandhi · Answer

Are imaginary roots not considered roots.the questions only mentions roots and not distinct roots. The answer is still (D).
Am i thinking right?

Posted from my mobile device

HoneyLemon · Answer

Dilpreetgandhi . if c <0 in the original equation . then the discriminant D >0 If D >0 , then there won't be imaginary roots , as per properties of quadratic equation. . ax^2+bx+c (If D = b^2 - 4ac >0 ,-- roots are real and distinct D =0 , -- roots are real and equal D < 0 ,-- roots are imaginary ) Hope it helps .

CAMANISHPARMAR · Answer

Although a very easy question but does not seem like a GMAT question as GMAT does not test your knowledge of discriminants.

What is the source of this question?

FridaElisa · Answer

I took from a essay in LBS web page.

In the equation x^2 − cx + c = 0, c is a constant, and x is a variable

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