In how many places does the graph of the equation y = ax^2 + bx + c in

Question

Competition Mode Question

In how many places does the graph of the equation  y = ax^2 + bx + c  intersect the x - axis?

(1)  ac = 4

(2)  \sqrt{b} = \sqrt{5}

GMATinsight · Answer

CONCEPT: Number of the intersection of a quadratic equation with X-axis = No. of Roots of the equation

To find whether roots are real, Non-real or Distinct we check the value of  b^2-4ac

If  b^2-4ac = 0  only one point of intersection because roots are real and equal
If  b^2-4ac < 0  ZERO point of intersection because roots are non-real
If  b^2-4ac > 0  Two point of intersection because roots are real and Different

Statement 1 : ac = 4

no information about b^2 hence 
NOT SUFFICIENT

Statement 2 : √b=√5

no information about ac hence 
NOT SUFFICIENT

Combining the statements

b = 5 an d ac = 4

i.e.  b^2-4ac = 5^2 - 4*4 = 9

i.e. Two points of intersection

Answer: Option C

CareerGeek · Answer

Question: How many x-intercepts does the graph have?
To find x-intercept(s), put  y = 0 
-->  ax^2 + bx + c = 0 
-->  x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}

Rule:
If  b^2 - 4ac > 0 , Number of intersections with x-axis = 2
If  b^2 - 4ac = 0 , Number of intersections with x-axis = 1
If  b^2 - 4ac < 0 , Number of intersections with x-axis = 0

(1)  ac = 4 
We do not know the value of  b . So, not possible to find values of ' x ' -->  Insufficient

(2)  √b = √5 
-->  b = 5 
We do not know the values of  a  &  c . So, not possible to find values of ' x ' -->  Insufficient

Combining (1) & (2),
 b^2 - 4ac = 5^2 - 4*4 = 9 (>0)  --> 2 intersections -->  Sufficient

Option C

freedom128 · Answer

y=ax^2+bx+c will intersect x-axis twice, if b^2-4ac>0
y=ax^2+bx+c will intersect x-axis once, if b^2-4ac=0
y=ax^2+bx+c will not intersect x-axis, if b^2-4ac<0

(1) ac=4
We don't know value of b. Thus, we cannot deduce how many times y=ax^2+bx+c intersects x-axis.
NOT SUFFICIENT

(2) b=5
We don't know value of a*c. Thus, we cannot deduce how many times y=ax^2+bx+c intersects x-axis.
NOT SUFFICIENT

(1)+(2) b^2-4ac = 5^2 - 4*4 = 9 >0
We can deduce that y=ax^2+bx+c intersects x-axis twice.
SUFFICIENT

FINAL ANSWER IS (C)

exc4libur · Answer

Find the discriminant
b^2-4ac>0: 2 solutions
b^2-4ac=0: 1 solutions
b^2-4ac<0: 1 solutions

(1) insufic 
 (2) insufic

√b=√5
√b≥0, so b≥0
b=5

(1/2) insufic

b^2-4ac
5^2-4(4)=25-16=9>0
2 solutions

Ans (C)

unraveled · Answer

In how many places does the graph of the equation  y=ax^2+bx+c  intersect the x - axis?

(1) ac = 4
For  y=ax^2+bx+c  
Discriminant =  b^2 - 4ac  which determines how many solution are there for the equation.
As nothing about b is given then discriminant is unknown. So we can't find the solution.

INSUFFICIENT.

(2) √b = √5
b = 5 nothing else.

INSUFFICIENT.

Together 1 and 2
We know for quadratic equations like  ax^2+bx+c ,  b^2−4ac  is called the discriminant

The points where the curve intersects with the x-axis are the solutions of the curve.
Discriminant helps us find all such points.

If  b^2 − 4ac > 0  curve will intersect x-axis at two points
If  b^2 − 4ac = 0  curve will intersect x-axis at one point
If  b^2 − 4ac < 0  curve will not intersect x-axis.

Here case I is applicable since  b^2 − 4ac = 9  i.e. 2 points.

SUFFICIENT.

Answer C.

minustark · Answer

To find at how many places the graph intersects x axis,we need the value of discriminate b^2 - 4ac.
Now, from 1) we know 4ac = 16. Three cases are possible: if b^2 > 16, the discriminant will be positive, in that case the equation has 2 solutions and it will intersect x axis at 2 points. But if b^2 < 16, the equation will have no real solutions and hence no x intercept. Again, if b^2 = 4ac , the equation will have just one solution. Insufficient

2) We don't know anything about the value of a and c. Insufficient.

Together, the discriminant can be determined. Sufficient.

C is the answer.

madgmat2019 · Answer

y=ax^2+bx+c

discernment is 0 = no solution = not touches x axis
discernment is + ve = 2 solutions = touches x axis at 2 points
discernment is - ve = no solution = not touches x axis

Discrimant value = b^2- 4ac

(1) ac=4........then b^2-16......
if b=4...then D=0
if b>=4 then D>0
if b<4 then D<0

so without D value we cannot the no.of points of intersection of y and x-axis

INSUFFICIENT

(2) √b=√5

same as above without the values of c.....its INSUFFICIENT

Combing both we can get the value of D=+ve.....
so there are 2 points of intersection
SUFFICIENT

OA:C

razorhell · Answer

Ans C

Check whether b^2 - 4ac is <0 or >0 or =0

(i)ac = 4 ..... then the discriminant depends on the values of b and the possible points at which the graph cuts x-axis can be 2,1 or 0. Not sufficient

(ii) b = 5 ..... then the discriminant depends on the values of a*c and the possible points at which the graph cuts x-axis can be 2,1 or 0. Not sufficient

Together b^2-4ac > 0 .... hence sufficnet

Ted2007 · Answer

To decide how many roots, we need to know whether b^2 - 4acis positive, 0, or negative, we can figure it out only by combining statement 1 and 2. Therefore it is C

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bumpbot · Answer

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In how many places does the graph of the equation y = ax^2 + bx + c in

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