Product of All Roots

The given function is a quadratic function. Hence, as what is taught to us at school [Vieta's formula], the formula for the product of the roots of a quadratic equation is $\frac {c}{a}$ , where $f(x)=a^2+bx+c$ . If you would notice, the given equation has 8 addends with each addends giving a quadratic equation with $x^2$ as its first term always. Thus, it would give us $8x^2$ as the first term of the simplified equation. Next is the last term of the simplified equation. We would be able to get the constant term of each addend by simply multiplying the constants of each addend with each other and adding them. This is equal to $(-1)(-2)+(-2)(-3)+(-3)(-4)+...+(-8)(-9) =240$ . Hence, the product of roots is equal to $\frac {c}{a} = \frac {240}{8} = 30$ . :D

[Latex edits by Calvin]

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As the constant in the quadratic equation represents the product of all the roots, therefore, by simplifying the equation, it became -

8x.x -[ 1+2+2+3+3+4+4+...+8+8+9]x +[1.2+2.3+3.4+4.5+...+8.9] = 0

Solving the constant term, and dividing by 8(a of quadratic equation) we get the answer to the question...

8x^2 - 80x + 240 = 0 \Rightarrow x^2 - 10x + 30 = 0 \Rightarrow (x-5)^2) - 25 + 30 = 0 \Rightarrow (x-5)^2 = -5 \Rightarrow x-5 = \pm \sqrt{-5} \Rightarrow x = 5 \pm \sqrt{5}i \Rightarrow (5 + \sqrt{5}i)(5 - \sqrt{5}i) = 25 + 5 = 30

Taking (x-2) common from first two expressions and (x-4) common from the next two and so on, we get (x-2)[(x-1) +(x-3)] + (x-4)[(x-3) + (x-5)] + (x-6)[(x-5) + (x-7)] + (x-8) [(x-7) + (x-9)] = 0 (x-2)(2x-4) + (x-4)(2x-8) + (x-6)(2x-12) + (x-8)(2x-16) =0 Taking 2 common 2(x-2)(x-2) +2 (x-4) (x-4) +2 (x-6) (x-6) +2(x-8)(x-8) =0 On division by 2 (x-2)^2 + (x-4)^2 + (x-6)^2 + (x-8)^2 =0 On expanding we get 4x^2- 40x + 120=0 On division by 4, we get x^2 -10x+30=0 By applying Vieta's theorem Product of roots of a quadratic equation=c/a,where c=constant term and a=coefficient of x^2 we get product of roots=30/1=30

coefficient of x^2=8
and coefficient of independent terms = 2+6+13+20+30+42+56+72=240 product of roots = 240/8 =30

sum of constant term =240 coefficient of x2=8 hence product of roots to the equation=30

f(x)=(x−1)(x−2)+(x−2)(x−3) +(x−3)(x−4)+(x−4)(x−5)+(x−5)(x−6) +(x−6)(x−7)+(x−7)(x−8)+(x−8)(x−9)=0 we see that the constant term in the equation is(-1)(-2)+(-2)(-3)+...+(-8)(-9) =240 Also,coefficient of x^2 in the equation is equal to the number of addends in the equation, as each addend has a coefficient of 1 for x^2. So,the coefficient of x^2 is 8

So product of roots=constant term/coefficient of x^2 =240/8 =30

The constant term of the equation is 240, and the leading coefficient is 8. According to Vieta's Formulas, the product of the roots of an even degreed polynomial is the constant over the leading coefficient, so 30 is your answer.

We have...

(x-1)(x-2) + (x-2)(x-3) + (x-3)(x-4) + (x-4)(x-5) + (x-5)(x-6) + (x-6)(x-7) + (x-7)(x-8) + (x-8)(x-9) = 8x^2 - 71x + 240

So, product of roots= 240/8= 30

The leading coefficient is 8. By Vieta’s formulae, we have that the product of all roots is given by the constant term divided by the leading coefficient. Hence the product of all roots is $\begin{aligned} \frac {f(0)}{8} & = \frac {1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7 + 7 \times 8 + 8 \times 9}{8} & \\ & = \frac{2 + 6 + 12 + 20 + 30 + 42 + 56 + 72}{8} & \\ & = \frac {240}{8}=30 &\\ \end{aligned}$