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One root of the quadratic equation ax^2+bx+c=0

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Director
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 726
Location: India
GPA: 3.64

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08 Feb 2018, 10:46
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Difficulty:

65% (hard)

Question Stats:

44% (01:10) correct 56% (03:43) wrong based on 34 sessions

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One root of the quadratic equation $$ax^2+bx+c=0$$ is 2 and one root of the quadratic equation $$cx^2+bx+a=0$$ is $$\frac{-1}{3}$$. What is the sum of the roots of the first equation?
A) $$\frac{5}{3}$$
B) -1
C) -2
D) -3
E) -5

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Joined: 25 Feb 2013
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08 Feb 2018, 11:14
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souvonik2k wrote:
One root of the quadratic equation $$ax^2+bx+c=0$$ is 2 and one root of the quadratic equation $$cx^2+bx+a=0$$ is $$\frac{-1}{3}$$. What is the sum of the roots of the first equation?
A) $$\frac{5}{3}$$
B) -1
C) -2
D) -3
E) -5

$$ax^2+bx+c=0$$ Sum of roots of this equation will be $$-\frac{b}{a}$$

as one root is $$2$$ so we have

$$4a+2b+c=0$$------------(1)

$$cx^2+bx+a=0$$ as one root is $$-\frac{1}{3}$$

we have $$\frac{c}{9}-\frac{b}{3}+a=0=>9a-3b+c=0$$-------(2)

subtract equation (1) from (2) we get $$5a-5b=0 => \frac{b}{a}=1$$

so $$-\frac{b}{a}=-1$$

Option B
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08 Feb 2018, 11:28
souvonik2k wrote:
One root of the quadratic equation $$ax^2+bx+c=0$$ is 2 and one root of the quadratic equation $$cx^2+bx+a=0$$ is $$\frac{-1}{3}$$. What is the sum of the roots of the first equation?
A) $$\frac{5}{3}$$
B) -1
C) -2
D) -3
E) -5

Let m be the second root of first equation and n be the second root of second equation

For First Equation
Sum of roots = -(b/a)
so, m+2 = -(b/a) ------------ (1)

Product of roots = c/a
so, 2m = c/a ---------------(2)

For Second Equation
Sum of roots = -(b/c)
so, m+2 = -(b/c) ------------ (3)

Product of roots = a/c
so, 2m = a/c ---------------(4)

Multiply (2) and (4) and we get m*n = -(3/2)

Multiply (2) and (3) and we get
-(b/a) = (2m)*(n-1/3)

Now -(b/a) = m+2

So, m+2 = (2m)(n-1/3)

we have value of mn so value of m can be obtained from above equation.

m = -3

Sum of roots of first equation = -3+2 = -1

Re: One root of the quadratic equation ax^2+bx+c=0 &nbs [#permalink] 08 Feb 2018, 11:28
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