Given the equation x^2 + bx + c = 0, where b and c are constants, what

Question

Given the equation x^2 + bx + c = 0, where b and c are constants, what is the value of c?

(1) The sum of the roots of the equation is zero.

(2) The sum of the squares of the roots of the equation is equal to 18.

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CareerGeek · Answer

Formula: For the equation,  ax^2 + bx + c = 0 , Sum of the roots = - \frac{b}{a}  & Product of the roots =  \frac{c}{a}

(1) The sum of the roots of the equation is zero.
--> - \frac{b}{1} = 0 
-->  b = 0 
No information about ' c ' -->  Insufficient

(2) The sum of the squares of the roots of the equation is equal to 18
Let the roots be ' m ' & ' n '
-->  m^2 + n^2 = 18 
-->  (m + n)^2 - 2mn = 18 
-->  (-b)^2 - 2*c = 18 
-->  b^2 - 2c = 18  -->  Insufficient

Combining (1) & (2),
 0^2 - 2c = 18 
-->  c = -9  -->  Sufficient

Option C

GMATinsight · Answer

Sum of the roots of equation ax^2=bx+c = 0, is given by -b/a
Product of the roots of equation ax^2=bx+c = 0, is given by -c/a

Statement 1: The sum of the roots of the equation is zero.

-b/1 = 0
i.e. b = 0

i.e. x^2 = -c

NOT SUFFICIENT

Statement 2 : The sum of the squares of the roots of the equation is equal to 18

It's unknown whether roots are integers or non-integers hence infinite solutions exist hence

NOT SUFFICIENT

COmbinig the two statements

Sum of roots = 0

i.e. if one root = p then other root = -p

Also, p^2+(-p)^2 = 18

i.e. p^2 = 9

i.e. p =  + 3

i.e. Product of the roots = (+3)*(-3) = -9 = c/1 (as per the rule mentioned above in text box)

Answer: Option C

MathRevolution · Answer

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. Visit https://www.mathrevolution.com/gmat/lesson for details. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary. Assume p and q are roots of the equation of x^2+bx+c = 0 . Then we have (x-p)(x-q) = x^2+bx+c = 0 . We have x^2 - (p+q)x + pq = x^2 + bx + c . Thus we have p+q = -b and pq = c . Since we have 4 variables ( p and q ) and 2 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first. Conditions 1) & 2) We have p+q = -b = 0 from condition 1). We have p^2 + q^2 = (p+q)^2 - 2pq = 0 - 2c = -2c = 18 . Thus we have c = -9 . Since both conditions together yield a unique solution, they are sufficient. Therefore, C is the answer. Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

CrackverbalGMAT · Answer

For a standard quadratic equation of the form a x^2  +bx+c = 0,
Sum of roots = -(b/a) and Product of roots = c/a.

Comparing the equation  x^2 +bx+c = 0 to the standard form, a =1, b = b and c=c.
Let the roots of the given equation be α and β. Then, α+β = -b and αβ = c.

From statement I alone, we can say that b = 0. This is insufficient to find out the value of c.
Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone,  α^2  +  β^2  = 18. This is insufficient to find unique values of α and β, hence insufficient to find a value for c since c = αβ. 
Statement II alone is insufficient. Answer option B can be eliminated. The possible answer options are C or E.

Combining both statements together, we have the following:
 From statement I alone, we know that α+β = -b = 0. 
From statement II alone, we know that  α^2  +  β^2  = 18
From the question data, we know that αβ = c.

In any question on Algebra, if you have the data about the sum and product of two variables along with the sum of their squares, clearly, the question wants you to apply the Algebraic identity of  (a+b)^2 .

(a+b)^2  =  a^2  +  b^2  + 2ab.

Applying this to our situation, we have  (α+β)^2  =  α^2 + β^2 +2αβ. Substituting the values, we have,

(0)^2  = 18 + 2c. Solving this gives us c=-9.
The combination of statements is sufficient. Answer option E can be eliminated.

The correct answer option is C .

Hope that helps!

bumpbot · Answer

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Given the equation x^2 + bx + c = 0, where b and c are constants, what

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