Final answer:
The cubic equation 2x³ -9x²+7x+6=0 can be potentially solved by finding rational roots and factoring, or if reducible to a quadratic form, by using the quadratic formula. Otherwise, numerical methods or advanced algebraic techniques might be necessary.
Step-by-step explanation:
To solve the cubic equation 2x³ -9x²+7x+6=0, unlike a quadratic equation, we may need to use alternative methods such as factoring, synthetic division, or numerical methods if an obvious factorization is not apparent. However, if the equation can be reduced to a quadratic or factored into linear terms, we can then find the solutions for x.
First, we should check for possible rational roots using the Rational Root Theorem. If we find a rational root, let's say 'p/q', we can use synthetic division or polynomial long division to factor it out, reducing the cubic equation to a quadratic one. Once we have a quadratic equation, we can use the quadratic formula:
x = √((-b ± √(b² - 4ac))/(2a))
Where 'a', 'b', and 'c' are the coefficients from the quadratic equation in the form ax²+bx+c=0.
Alternatively, if synthetic division or factoring do not simplify the equation, we may need to employ numerical methods or more advanced algebraic techniques like Cardano's formula to find the real solutions to the cubic equation.