What is the value of K for which the roots of the quadratic equation 5

Question

What is the value of K for which the roots of the quadratic equation  5x^2 + 13x + K = 0  are reciprocals to each other?

A. -5
B. -3
C. 1
D. 3
E. 5

GMATinsight · Accepted Answer

For a quadratic equation  ax^2+bx+c = 0 ,
The product of Roots  α*β= \frac{c}{a}

Comparing  5x^2 + 13x + K = 0  with  ax^2+bx+c = 0 ,

a = 5, b = 13, c = K

Given,  α = \frac{1}{β}

i.e.  α*β = 1

But,  α*β = \frac{c}{a} = \frac{K}{5}

Therefore,  α*β = \frac{K}{5} = 1

i.e.   K = 5

Answer: Option E

MarmikUpadhyay · Answer

Given equation:  5x^2 + 13x + K = 0

Roots of the equation by Quadratic formula:
x =  \frac{(-13 + \sqrt{13 * 13 - 4 * 5 * K})}{(2 * 5)} 
and x =  \frac{(-13 - \sqrt{13 * 13 - 4 * 5 * K})}{(2 * 5)}

Both roots are reciprocal of each other:
=>  \frac{(-13 + \sqrt{169 - 20 * K})}{(10)}  =  \frac{(10)}{(-13 - \sqrt{169 - 20 * K})}

=> 10 * 10 = (-13 +  \sqrt{169 - 20 * K} ) * (-13 -  \sqrt{169 - 20 * K} ) 
Using  (a - b) * (a + b) = a^2 - b^2  ]
100 = (-13) * (-13) - (169 -20 * K) 
=> 100 = 169 - (169 - 20 * K) = 20 * K
=> K = 100/20 = 5

Alternatively,
Product of roots =  \frac{K}{5}  
Roots are reciprocal of each other, so product of roots = 1
=>  \frac{K}{5}  = 1 
=> K = 5

So, correct answer is option  E .

TarunKumar1234 · Answer

The quadratic equation 5x^2+13x+K=0 And, Product of roots = m*1/m = k/5 (If m= one of the roots of the equation then another root will be 1/m) or, k= 5. So, I think E.

winterschool · Answer

What is the value of K for which the roots of the quadratic equation 5x2+13x+K=05x2+13x+K=0 are reciprocals to each other?

A. -5
B. -3
C. 1
D. 3
E. 5

(x+a) * (x-1/a) = 0
x^2 + (a+1/a) * x + 1 = 0

given, 5x^2 + 13*x + k = 0 or, x^2 + 13/5 x + k/5 = 0

a+1/a = 13/5 and 1 = k/5

so, k = 5

Answer E

Ae95 · Answer

I am very lost with this question

GMATinsight · Answer

HI  Ae95

To understand such questions, you need to be familiar with terminology such as Roots, Quadratic equation which is basic of Algebra unit

Then you should know how to factorise a quadratic equation

Simultaneously, you should know Sri Dharacharya (Discriminant) formula and

At last, you should know the properties of Sum and product of the roots (the derivation is in first three steps)

In case you are missing something specific then please share and I will try my bets to share with you the required supplement.

What is the value of K for which the roots of the quadratic equation 5

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What is the value of K for which the roots of the quadratic equation 5

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