System of Cubic Equations

Algebra Level 5

If distinct real numbers $a$ , $b$ and $c$ satisfy the system of equations $a^3-3a^2 +2a-33=a^2+b^2+c^2,\\ b^3-3b^2+2b-33=a^2+b^2+c^2,\\ c^3-3c^2+2c-33=a^2+b^2+c^2,$ what is the value of $a \times b \times c$ ?

36 33 40 38

2 solutions

Led Tasso
Jul 28, 2014

Consider the cubic equation $x^3 - 3x^2 + 2x - 33 - a^2 - b^2 - c^2 = 0$ . According to question, $a,b,c$ are three roots of this equation. We have to find $abc$ which, according to Vieta's formulae is equal to $33 + a^2 + b^2 + c^2$ .

Two other relations from Vieta's rule are,

$a + b + c = 3$ and $ab + bc + ca = 2$ .

Hence $abc = 33 + a^2 + b^2 + c^2 = 33 + (a + b + c)^2 -2(ab + bc + ca) = 33 + 3^2 -2(2) = 33 + 9 - 4$ $=$ $\boxed{38}$

Definitely the best of all.

Nishant Sharma - 6 years, 10 months ago

Ravi R
Mar 1, 2014

Factorising the options and reverse substituting is one way. Please let me know how to solve this.

assume d equation to be: x^3 - 3x^2 + 3x - 33 =k Den get the value of k by a^2 + b^2 + c^2

Sandesh Kumar - 7 years, 2 months ago

@Ravi R , view my solution if it helps.

Led Tasso - 6 years, 10 months ago

At least you can narrow it down to a most likely two choices.

Robert Fritz - 7 years, 3 months ago

System of Cubic Equations

2 solutions

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