Sum of solutions

Algebra Level 2

What is the sum of all possible solutions for x x of the equation x ( x − k ) = k + 1 x ( x - k ) = k + 1 for constant k k ?

2 k − 1 2k-1 k − 1 k-1 k k − 1 -1 1 1 None of these choices 0 0 k + 1 k+1

Abhyudaya Apoorva by Abhyudaya Apoorva , 27, India

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2 solutions

Abhyudaya Apoorva
Dec 28, 2016

Any quadratic equations in x with two roots a and b can be expressed as, (x - a)(x - b) = 0. Which on expansion takes the form, x² - (a + b)x - ab = 0

Hence, the sum of the roots = -(coefficient of x in the expanded form)

Now, x(x - k) = k + 1 => x² - kx - (k + 1) = 0

Hence, sum of the roots = -(-k) = k

3 Helpful 0 Interesting 0 Brilliant 0 Confused

The equation can be rewritten as:

x 2 − k x − k − 1 = 0 x^2-kx-k-1=0

According to Vieta's formula, the sum of roots to this equation is equal to − ( − k ) 1 = k \dfrac{-(-k)}{1}=k .

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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