Question

To find the approximate real solutions of x3−2x=3x2−1, you can rearrange the equation to x3−3x2+2x−1=0 and then solve for x. It's a cubic equation, so you can use numerical methods or calculators to find the solutions. The solutions should be rounded to the nearest hundredth.

Answer 1

The student is seeking to solve a cubic equation by converting it to standard form and then applying numerical methods to find the real solutions. The quadratic formula can be used for quadratic equations, but cubic equations require different techniques such as graphing or iteration to find their solutions.

Step-by-step explanation:

The subject's question is about finding the approximate real solutions to a cubic equation, x3 - 2x = 3x2 - 1, by rearranging it to x3 - 3x2 + 2x - 1 = 0 and then solving for x. The student correctly suggests that numerical methods or calculators can be used to find solutions to cubic equations. When dealing with a cubic equation, one does not have a formula as simple as the quadratic formula, which is used for quadratic equations of the form ax2 + bx + c = 0. However, methods such as graphing, iteration, or software that can approximate roots may be used to find the solutions.

To solve a quadratic equation, the quadratic formula, x = (-b ± √(b2 - 4ac)) / (2a), is applied where a, b, and c are coefficients from the equation ax2 + bx + c = 0. This formula allows for finding both real and complex solutions of the quadratic equation, depending on the discriminant value (b2 - 4ac).