The system of equations are \(\boxed{y = 12{x^3} - 5x}{\text{ and }}\boxed{y = 2{x^2} + x + 6}\) that can be used to find the roots of the equation \(12{x^3} - 5x = 2{x^2} + x + 6.\) Option (a) is correct.
Further explanation:
Given:
The equation is \(12{x^3} - 5x = 2{x^2} + x + 6.\)
The options are as follows,
(a). \(y = 12{x^3} - 5x{\text{ and }}y = 2{x^2} + x + 6\)
(b). \(y = 12{x^3} - 5x + 6{\text{ and }}y = 2{x^2} + x\)
(c). \(y = 12{x^3} + 2{x^2} - 4x{\text{ and }}y = 6\)
(d). \(y = 12{x^3} + 2{x^2} - 4x + 6{\text{ and }}y = 0\)
Explanation:
The given equation is \(12{x^3} - 5x = 2{x^2} + x + 6.\)
Consider the left hand side of the equation \(12{x^3} - 5x = 2{x^2} + x + 6\) as y and the right hand side of the equation \(12{x^3} - 5x = 2{x^2} + x + 6\) as y.
\(\begin{aligned}y&= 12{x^3} - 5x\\y &= 2{x^2} + x + 6 \\ \end{aligned}\)
The equation matches with option (a). Hence, option (a) is correct.
The system of equations are \(\boxed{y = 12{x^3} - 5x}{\text{ and }}\boxed{y = 2{x^2} + x + 6}\) that can be used to find the roots of the equation \(12{x^3} - 5x = 2{x^2} + x + 6\). Option (a) is correct.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Polynomials
Keywords: polynomial, solution, linear equation, quadratic equation, system of equations, solution of the equations.