Final answer:
The equation x⁹ − 5x³ + 6 = 0 is quadratic in form because it can be re-expressed as a standard quadratic equation using a substitution, such as u = x³. Once in this form, it can be solved by methods applicable to quadratic equations.
Step-by-step explanation:
Is the equation x⁹ − 5x³ + 6 = 0 quadratic in form? To determine this, we need to identify whether it can be re-expressed in the standard quadratic form ax²+bx+c=0, where a, b, and c are constants, and x represents the variable.
A quadratic in form is an equation that can be manipulated so that it takes the shape of a quadratic equation by using a substitution of variables. For the equation x⁹ − 5x³ + 6 = 0, we can make a substitution such as u = x³. Substituting, we get u² - 5u + 6 = 0, which is indeed a quadratic equation. Therefore, we can conclude that the original equation is quadratic in form and can be solved using techniques applicable to quadratic equations, such as factoring, completing the square, or using the quadratic formula.
It is crucial to understand how to recognize when an equation is quadratic in form, as it opens up a toolkit of strategies for finding solutions to the equation. The key is looking for an exponent that is a multiple of 2 and checking if a substitution can transform the equation into the standard quadratic form ax²+bx+c=0. Once in this form, solving the equation becomes much easier, providing valuable solutions of quadratic equations.