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If 2 + √3 is one root of the equation, [m]x^2 - ax + b = 0[/m], ......

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If 2 + √3 is one root of the equation, [m]x^2 - ax + b = 0[/m], ......  [#permalink]

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New post Updated on: 26 Jun 2019, 03:42
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A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

56% (01:46) correct 44% (02:10) wrong based on 66 sessions

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Originally posted by EgmatQuantExpert on 19 Jun 2019, 06:44.
Last edited by EgmatQuantExpert on 26 Jun 2019, 03:42, edited 1 time in total.
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Re: If 2 + √3 is one root of the equation, [m]x^2 - ax + b = 0[/m], ......  [#permalink]

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New post 19 Jun 2019, 11:30
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If one root of the quadratic equation is \(2+\sqrt{3},\)the other root is \(2-\sqrt{3}.\)Now, sum of roots = a = \(2+\sqrt{3}+2-\sqrt{3}\) = 4
product of roots = b = \((2+\sqrt{3})(2-\sqrt{3})\)= \(2^2 -{\sqrt{3}}^2\) = 4-3 = 1
Hence, a + b = 4 + 1 = 5 (Option E)
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Re: If 2 + √3 is one root of the equation, [m]x^2 - ax + b = 0[/m], ......  [#permalink]

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New post 19 Jun 2019, 11:36
giving a try
\(x^2 - ax + b = 0\),
so
we have ; (2+√3)^2- (2+√3)a+b=0
now value of a,b has to be such that eqn is 0
so let a= 4 and b=1
we get
7+4√3-8-4√3+1=0
sufficient
a+b = 4+1 ; 5
IMO E

EgmatQuantExpert wrote:
If 2 + √3 is one root of the equation, \(x^2 - ax + b = 0\), where a, b are positive numbers, then what is the value of a + b?

    (A) 0
    (B) 1
    (C) 2√3
    (D) 4
    (E) 5

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Re: If 2 + √3 is one root of the equation, [m]x^2 - ax + b = 0[/m], ......  [#permalink]

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New post 26 Jun 2019, 03:44

Solution


Given:
    • 2 + √3 is one root of the quadratic equation, \(x^2 - ax + b = 0\)
    • a, b are positive numbers

To find:
    • The value of a + b

Approach and Working Out:
    • We are given that 2 + √3 is one root of the quadratic equation, \(x^2 - ax + b = 0\)
    • And, we know that, if a +√ b is one root of a quadratic equation, then the other root will be a - √b.
      o Thus, 2 + √3, and 2 - √3 are the two roots

    • Now, from the given quadratic equation, \(x^2 - ax + b = 0\)
      o Sum of the roots = -(-a) = a = (2 + √3) + (2 - √3) = 4
      o Product of the roots = b = (2 + √3) * (2 - √3) = 4 – 3 = 1

    • Therefore, a + b = 4 + 1 = 5

Hence, the correct answer is Option E.

Answer: E


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Re: If 2 + √3 is one root of the equation, [m]x^2 - ax + b = 0[/m], ......  [#permalink]

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New post 26 Jun 2019, 04:53
EgmatQuantExpert wrote:
If 2 + √3 is one root of the equation, \(x^2 - ax + b = 0\), where a, b are positive numbers, then what is the value of a + b?

    (A) 0
    (B) 1
    (C) 2√3
    (D) 4
    (E) 5

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If 1 root is x + sqrt(y)
The other root will be x - sqrt(y)

So other root is 2 - 3^1/2

a = sum of roots = 4
b = product of roots = 4 - 3 = 1

a + b = 5

Option E

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Re: If 2 + √3 is one root of the equation, [m]x^2 - ax + b = 0[/m], ......   [#permalink] 26 Jun 2019, 04:53
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