At how many points does the graph of a quadratic equation cut x axis

Question

At how many points does the graph of a quadratic equation cut x axis 1) Sum of the roots is 5 2) Product of the roots is 6 Souce: www.GMATinsight.com

niks18 · Accepted Answer

Quadratic equation is of the form  y=ax^2+bx+c . at  x  intercept  y=0

hence  ax^2+bx+c=0 => x^2+\frac{b}{a}x+\frac{c}{a}=0

we need to find the value of  \frac{b}{a}  &  \frac{c}{a}  to arrive at the equation and hence at the  x -intercepts

Statement 1 : From Vieta's formula , sum of roots =-\frac{b}{a}=5 , hence  \frac{b}{a}=-5  but nothing mentioned about  \frac{c}{a} .  Insufficient

Statement 2 : Again from Vieta's formulas, product of roots =\frac{c}{a}=6 . but nothing mentioned about  \frac{b}{a} .  Insufficient

Combining 1 & 2 we get all the required values, hence our equation is  x^2-5x+6=0

Find the roots of this equation to know the  x -intercepts

x^2-5x+6=(x-3)(x-2)=0 . therefore  x -intercepts are  3  and  2 .  Sufficient

Option  C

kawal27 · Answer

1) Sum of the roots is 5 2+3=5 1+4=5 insufficient
2) Product of the roots is 6 2*3=6 1*6=6 insufficient

combined : x-intercepts are 3 and 2 Sufficient.

ans : C

giovannisumano · Answer

This question asks the number of roots of the equation. Visualize a quadratic equation in a plane.

There are 3 possibilities for a quadratic equation:
1. It has no roots
2. It has 1 root (when the vertex touches the x axis)
3. It has 2 roots

Statement 1 tells us that the equation has at least 1 root, but we don't know whether it has one of two --> Insufficient 
Statement 2 tells gives us the same information -->  Insufficient

Statements 1+2 tell us that the equation has to have 2 roots, because if it had only 1 root then statements 1 and 2 would give us the same value (if there was only one root its sum and product would be the same) -->  Sufficient

js1986 · Answer

From Statement 2, cant we say that it will intersect the X-axis at 2 points? As it cannot have only 1 root (6 cannot be written as a product of 2 numbers which are the same except Square root of 6 but a quadratic equation cannot have square root of 6 as the only root as if roots are irrational it must be conjugates)?

At how many points does the graph of a quadratic equation cut x axis

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At how many points does the graph of a quadratic equation cut x axis

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