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(A) x^2-2*x+A is positive for all x (B) A*x^2+1 is positive for all x

(A) x^2- 2x+A is an upward parabola( coz the coefficient of x^2 is 1 > 0) . The statement indeed means that the quadratic equation has no roots. In that case, the parabola totally belong to the upper half of the coordination plane without touching the x-axis ( coz in case, it touches/ is tangent to x-axis, the value of the expression is 0- not positive!)
--> the discriminant of the equation <0 ( means "no roots at all")
--> (-2)^2 - 4A <0 --> A>1> 0 --> answer to the question : YES
--> suff

(B).
Solve this equation Ax^2+1 in the same manner to statement 1's. We have A>0 make the expression positive always --> Answer to the question: YES.
However:
In case A= 0 --> 0*x^2 +1 = 1 > 0 --> A= 0 is okie
--> A>=0 is okie --> we can only say that A is not negative but can't conclude that A must be positive or not ( due to the case A=0 )
--> insuff.

(A) x^2- 2x+A is an upward parabola( coz the coefficient of x^2 is 1 > 0) . The statement indeed means that the quadratic equation has no roots. In that case, the parabola totally belong to the upper half of the coordination plane without touching the x-axis ( coz in case, it touches/ is tangent to x-axis, the value of the expression is 0- not positive!) --> the discriminant of the equation <0 ( means "no roots at all") --> (-2)^2 - 4A <0 --> A>1> 0 --> answer to the question : YES --> suff. I go for A.

Laxi, i get E here. suppose x = -1, and A = -1 or 1, the equation x^2-2*x+A holds true.

if x = -1 and A = -1.
x^2-2*x+A = 1 +2 -1=2

if x = -1 and A = 1.
x^2-2*x+A = 1 +2 +1=4

i know i might be missing something. please elabroate tmore...............

(A) x^2-2*x+A is positive for all x (B) A*x^2+1 is positive for all x

(A) x^2- 2x+A is an upward parabola( coz the coefficient of x^2 is 1 > 0) . The statement indeed means that the quadratic equation has no roots. In that case, the parabola totally belong to the upper half of the coordination plane without touching the x-axis ( coz in case, it touches/ is tangent to x-axis, the value of the expression is 0- not positive!) --> the discriminant of the equation <0 ( means "no roots at all") --> (-2)^2 - 4A <0 --> A>1> 0 --> answer to the question : YES --> suff

(B). Solve this equation Ax^2+1 in the same manner to statement 1's. We have A>0 make the expression positive always --> Answer to the question: YES. However: In case A= 0 --> 0*x^2 +1 = 1 > 0 --> A= 0 is okie --> A>=0 is okie --> we can only say that A is not negative but can't conclude that A must be positive or not ( due to the case A=0 ) --> insuff.

I go for A.

Its E

1) Take x = 10 then the equation comes out to be 100-20 + A > 0 i.e A > -80. So A can either be +ve or -ve - INSUFF

2) Take x = 10 then 10A > -1 i.e A > -1/10. So A can either be +ve or -ve - INSUFF

Combining both of them also yields same result. _________________

(A) x^2-2*x+A is positive for all x (B) A*x^2+1 is positive for all x

(A) x^2- 2x+A is an upward parabola( coz the coefficient of x^2 is 1 > 0) . The statement indeed means that the quadratic equation has no roots. In that case, the parabola totally belong to the upper half of the coordination plane without touching the x-axis ( coz in case, it touches/ is tangent to x-axis, the value of the expression is 0- not positive!) --> the discriminant of the equation <0 ( means "no roots at all") --> (-2)^2 - 4A <0 --> A>1> 0 --> answer to the question : YES --> suff

(B). Solve this equation Ax^2+1 in the same manner to statement 1's. We have A>0 make the expression positive always --> Answer to the question: YES. However: In case A= 0 --> 0*x^2 +1 = 1 > 0 --> A= 0 is okie --> A>=0 is okie --> we can only say that A is not negative but can't conclude that A must be positive or not ( due to the case A=0 ) --> insuff.

I go for A.

Its E

1) Take x = 10 then the equation comes out to be 100-20 + A > 0 i.e A > -80. So A can either be +ve or -ve - INSUFF

2) Take x = 10 then 10A > -1 i.e A > -1/10. So A can either be +ve or -ve - INSUFF

Combining both of them also yields same result.

you're ignoring the part "for all x"

"for all x" means this value of A makes all the values of x make the expressions >0

Laxie is right. The critical point is x=0. It is A. B is insufficient since +1 displaces the parabola upwards making it impossible for us to determine if A is postitive.

(A) x^2-2*x+A is positive for all x (B) A*x^2+1 is positive for all x

(A) x^2- 2x+A is an upward parabola( coz the coefficient of x^2 is 1 > 0) . The statement indeed means that the quadratic equation has no roots. In that case, the parabola totally belong to the upper half of the coordination plane without touching the x-axis ( coz in case, it touches/ is tangent to x-axis, the value of the expression is 0- not positive!) --> the discriminant of the equation <0 ( means "no roots at all") --> (-2)^2 - 4A <0 --> A>1> 0 --> answer to the question : YES --> suff

(B). Solve this equation Ax^2+1 in the same manner to statement 1's. We have A>0 make the expression positive always --> Answer to the question: YES. However: In case A= 0 --> 0*x^2 +1 = 1 > 0 --> A= 0 is okie --> A>=0 is okie --> we can only say that A is not negative but can't conclude that A must be positive or not ( due to the case A=0 ) --> insuff.

I go for A.

Nice approach. Agree with A.

y = x^2- 2x+A

S1 says y > 0 for all x. only possible if this parabola doesn't touch x axis(is above x axis). rest follows. _________________

Whether you think you can or think you can't. You're right! - Henry Ford (1863 - 1947)

(A) x^2-2*x+A is positive for all x (B) A*x^2+1 is positive for all x

(A) x^2- 2x+A is an upward parabola( coz the coefficient of x^2 is 1 > 0) . The statement indeed means that the quadratic equation has no roots. In that case, the parabola totally belong to the upper half of the coordination plane without touching the x-axis ( coz in case, it touches/ is tangent to x-axis, the value of the expression is 0- not positive!) --> the discriminant of the equation <0 ( means "no roots at all") --> (-2)^2 - 4A <0 --> A>1> 0 --> answer to the question : YES --> suff

(B). Solve this equation Ax^2+1 in the same manner to statement 1's. We have A>0 make the expression positive always --> Answer to the question: YES. However: In case A= 0 --> 0*x^2 +1 = 1 > 0 --> A= 0 is okie --> A>=0 is okie --> we can only say that A is not negative but can't conclude that A must be positive or not ( due to the case A=0 ) --> insuff.

To those who are supporting A, can you please clarify this: The first statement x^2-2x+A is positive for all values

lets take x=-2; this implies that A+8>0 Where A can range from -7 to +infinity........

What you misunderstand here is "for all x"
We CAN'T take a certain value of x to test for A. What we should know is that the satisfactory values of A must be those which make x^2-2x+A positive for ALL x

In your example, by plugging -2 to x , you concluded that A is from -7 to infinity. Then, I pick a certain A in this range, for example A = 0
--> the expression becomes: x^2 - 2x
with x= 1, the expression = -1 < 0 ---> This A (=0) can't satisfy the first statement.

Laxieqv,
Please help me in understanding the basis for assuming that Discriminant D < 0.
TIA.

laxieqv wrote:

HIMALAYA wrote:

Is A positive?

(A) x^2-2*x+A is positive for all x (B) A*x^2+1 is positive for all x

(A) x^2- 2x+A is an upward parabola( coz the coefficient of x^2 is 1 > 0) . The statement indeed means that the quadratic equation has no roots. In that case, the parabola totally belong to the upper half of the coordination plane without touching the x-axis ( coz in case, it touches/ is tangent to x-axis, the value of the expression is 0- not positive!) --> the discriminant of the equation <0 ( means "no roots at all") --> (-2)^2 - 4A <0 --> A>1> 0 --> answer to the question : YES --> suff

(B). Solve this equation Ax^2+1 in the same manner to statement 1's. We have A>0 make the expression positive always --> Answer to the question: YES. However: In case A= 0 --> 0*x^2 +1 = 1 > 0 --> A= 0 is okie --> A>=0 is okie --> we can only say that A is not negative but can't conclude that A must be positive or not ( due to the case A=0 ) --> insuff.

Laxieqv, Please help me in understanding the basis for assuming that Discriminant D < 0. TIA.

laxieqv wrote:

HIMALAYA wrote:

Is A positive?

(A) x^2-2*x+A is positive for all x (B) A*x^2+1 is positive for all x

(A) x^2- 2x+A is an upward parabola( coz the coefficient of x^2 is 1 > 0) . The statement indeed means that the quadratic equation has no roots. In that case, the parabola totally belong to the upper half of the coordination plane without touching the x-axis ( coz in case, it touches/ is tangent to x-axis, the value of the expression is 0- not positive!) --> the discriminant of the equation <0 ( means "no roots at all") --> (-2)^2 - 4A <0 --> A>1> 0 --> answer to the question : YES --> suff

(B). Solve this equation Ax^2+1 in the same manner to statement 1's. We have A>0 make the expression positive always --> Answer to the question: YES. However: In case A= 0 --> 0*x^2 +1 = 1 > 0 --> A= 0 is okie --> A>=0 is okie --> we can only say that A is not negative but can't conclude that A must be positive or not ( due to the case A=0 ) --> insuff.

I go for A.

The easy way to solve this problem is to picture the graph of the expression, an UPWARD parabola it is.
If the parabola intersects or touches the x-axis, in other words, the equation x^2-2x+A= 0 has roots --> PART of the parabola lies below the x-axis ----> that PART of the expression is SMALLER or EQUAL to 0 ( that means "not positive" )
=> So, in order to make the expression >0 for ALL x, the equation x^2-2x+A MUST have no roots
---> the condition for that to happen is that the discriminant D <0.

For my Cambridge essay I have to write down by short and long term career objectives as a part of the personal statement. Easy enough I said, done it...