Indirect substitution

Algebra Level 3

( 13 x − x 2 x + 1 ) ( x + 13 − x x + 1 ) = 42 \left(\frac {13x-x^2}{x+1}\right) \left(x + \frac {13-x}{x+1}\right) = 42

If the roots of the equation above are a a , b b , c c , and d d , find 2020 ( a + b + c + d ) 2020(a+b+c+d) .


The answer is 26260.

Nitin Kumar by Nitin Kumar , 14, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

3 solutions

Chew-Seong Cheong
Jan 10, 2020

( 13 x − x 2 x + 1 ) ( x + 13 − x x + 1 ) = 4 ( 13 x − x 2 x + 1 ) ( x ( x + 1 ) + 13 − x x + 1 ) = 4 ( 13 x − x 2 ) ( x 2 + 13 ) = 4 ( x + 1 ) 2 169 x − 13 x 2 + 13 x 3 − x 4 = 4 x 2 + 8 x + 4 ⟹ x 4 − 13 x 3 + 17 x 2 − 161 x + 4 = 0 \begin{aligned} \left(\frac {13x-x^2}{x+1}\right) \left(x + \frac {13-x}{x+1}\right) & = 4 \\ \left(\frac {13x-x^2}{x+1}\right) \left(\frac {x(x+1)+13-x}{x+1}\right) & = 4 \\ (13x-x^2)(x^2+13) & = 4(x+1)^2 \\ 169x - 13x^2 + 13x^3 - x^4 & = 4x^2 + 8x + 4 \\ \implies x^4 \red{- 13}x^3+17x^2 - 161x + 4 & = 0 \end{aligned}

By Vieta's formula , the sum of roots a + b + c + d = − ( − 13 ) = 13 ⟹ 2020 ( a + b + c + d ) = 26260 a+b+c+d = -(\red{-13}) = 13 \implies 2020 (a+b+c+d) = \boxed{26260} .

5 Helpful 3 Interesting 6 Brilliant 0 Confused

@Chew-Seong Cheong Yes, I think this is the most straightforward solution. :)

Jaydee Lucero - 1 year, 4 months ago

Log in to reply

Glad that you like it.

Chew-Seong Cheong - 1 year, 4 months ago

RHS is not 4 its 42

om prakash sharma - 1 year, 3 months ago

Simplifying the given equation we can write x 4 − 13 x 3 + 17 x 2 − 161 x + 4 = 0 x^4-13x^3+17x^2-161x+4=0 . Sum of the roots of the equation is 13 13 . So the required answer is 2020 × 13 = 26260 2020\times {13}=\boxed {26260}

0 Helpful 0 Interesting 2 Brilliant 1 Confused
Nitin Kumar
Jan 10, 2020

Let y=13-x/x+1
So xy(x+y)=42.
But xy+(x+y)=13.
Solving for xy and x+y, we get
X+y=6 and xy=7.
Hence, x=1,6,3+√2,3-√2 giving sum of roots=13




2 Helpful 0 Interesting 0 Brilliant 2 Confused

Did you mean to have the equation set equal to 42 42 ? The problem says 4 4 , but you used 42 42 in your solution and that would make your four roots correct.

The odd thing is, this typo changes the value of the four roots, but not their sum. So people are still getting the problem right.

Matthew Feig - 1 year, 5 months ago

Log in to reply

I changed the problem

Nitin Kumar - 1 year, 3 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...