For integers x, b and c, x^2 + bx + c = 0. Is x > 0?

Vyshak

Can you help me explain, how x1+x2 is negative ? is it because of -b/a ? with b as positive and a =1 ?

Hi Shrey9

yes you are correct. Sum of roots is -b/a so if b is positive and in this equation a=1, -b has to be negative

Comparing \(x^2\) + bx + c =0 with the general form of Quadratic Eqn \(ax^2 + bx + c = 0\)

We get a =1, b = b , and c = c

Let \(x_1\) and \(x_2\) be the two roots of the equation

Sum of the roots = \(x_1 + x_2\) =\(\frac{-b}{a}\) = \(\frac{-b}{1}\)= -b

Product of the roots = \(x_1 * x_2\) = \(\frac{c}{a}\) = \(\frac{c}{1}\) = c

St 1 :

b > 0, i.e -b < 0

\(x_1 + x_2 < 0\)

Let us plug in some values

Case 1 : \(x_1 = - 3, x_2 = - 2\) Both negative values

Case 2 : \(x_1 = - 3, x_2 = 2\) One value positive, one value negative

Not Sufficient

St 2 :

c > 0

\(x_1 * x_2 > 0\)

\(x_1 and x_2\) have the same sign

Both positive or Both negative

Not Sufficient

St 1 and St 2 Combined

Only Both negative remains

Sufficient

Target question: Is x > 0?

Statement 1: b > 0
Let's TEST some cases that satisfy statement 1:
If b = 5, and c = -14, the equation becomes x² + 5x - 14 = 0. Factor to get: (x + 7)(x - 2), which means EITHER x = -7 OR x = 2
If x = -7, then the answer to the target question is NO, x is NOT greater than 0
If x = 2, then the answer to the target question is YES, x is greater than 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: c > 0
Let's TEST some cases that satisfy statement 2:
Case a: b = -6, and c = 9. The equation becomes x² - 6x + 9 = 0. Factor to get: (x - 3)(x - 3), which means x = 3. So, the answer to the target question is YES, x is greater than 0
Case b: b = 6, and c = 9. The equation becomes x² + 6x + 9 = 0. Factor to get: (x + 3)(x + 3), which means x = -3. So, the answer to the target question is NO, x is NOT greater than 0

Statements 1 and 2 combined
We have: x² + bx + c = 0
Let's say that, when we factor x² + bx + c we get: x² + bx + c = (x + j)(x + k)
If c is POSITIVE, we know that EITHER j and k are both positive, OR j and k are both negative

We also know that, since b is POSITIVE, the sum of j and k must be POSITIVE

This allows us to rule out the possibility that j and k are both negative, since two negative numbers cannot add up to be positive.

So, it MUST be the case that j and k are both positive
In other words, x² + bx + c = (x + positive)(x + positive)
If (x + positive)(x + positive) = 0, then we can see that two solutions will both be negative.
For example, if (x + 2)(x + 3) = 0, then the two solutions are x = -2 and x = -3
Similarly, if (x + 1)(x + 10) = 0, then the two solutions are x = -1 and x = -10
If the two solutions will both be NEGATIVE, the answer to the target question is NO, x is NOT greater than 0
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

Step 1: Analyse Question Stem

x, b and c are integers.

A quadratic equation, \(x^2\) + bx + c = 0 is given.

We have to find out if x is more than zero. Since x represents the root of the quadratic equation, we have to find out if the root/s of the equation is/are more than zero.

Sum of roots of a quadratic equation = - \(\frac{b }{ a}\)

Product of roots of a quadratic equation = \(\frac{c }{ a}\)

For the given quadratic equation,
Sum of roots = - b and Product of roots = c

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: b > 0

Since b > 0, sum of roots = - (positive value) = a negative value

This means that either both roots are negative OR the bigger of the two roots is negative, while the smaller one is positive.

The data in statement 1 is insufficient to answer the question with a definite YES or NO.
Statement 1 alone is insufficient. Answer options A and D can be eliminated.

Statement 2: c > 0

Since c > 0, product of roots = a positive value
This means that either both roots are negative OR both roots are positive.

The data in statement 2 is insufficient to answer the question with a definite YES or NO.
Statement 2 alone is insufficient. Answer option B can be eliminated.

Step 3: Analyse Statements by combining

From statement 1: sum of roots = - (positive value) = a negative value

From statement 2: product of roots = a positive value

Both the constraints above can be satisfied only if both the roots are negative.

Is x > 0? NO.

The combination of statements is sufficient to answer the question with a definite NO.
Statements 1 and 2 together are sufficient. Answer option E can be eliminated.

The correct answer option is C.

x^2+bx+c=0
x= (-b+-Sqrt b^2-4c)/2
Is X>0'
-b+-Sqrt b^2-4c>0
a . Assume b>0
+-Sqrt b^2-4c>b
b^2-4c>b^2
c<0
b. Assume b<0
b^2-4c<b^2
c>0
From St .2 ,we know C>0
Therefore using both statement unique solution can be found.
Answer C

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