- #1
anemone
Gold Member
MHB
POTW Director
- 3,883
- 115
Show $20a^2+20b^2+5c^2\ge 64$ if $y=x^4+ax^3+bx^2+cx+4$ has a real root.
A quartic equation is a polynomial equation of degree 4, meaning that the highest exponent of the variable in the equation is 4. It is written in the form ax4 + bx3 + cx2 + dx + e = 0, where a, b, c, d, and e are constants and x is the variable.
A real root of a quartic equation is a value of x that makes the equation equal to 0 when substituted in. It is a solution that lies on the real number line, as opposed to imaginary roots which lie on the complex number plane.
The condition for a quartic equation to have a real root is that the discriminant of the equation, b2 - 4ac, must be greater than or equal to 0. This ensures that the equation has at least one real solution.
You can determine if a quartic equation has a real root by calculating the discriminant and checking if it is greater than or equal to 0. If it is, then the equation has at least one real solution. You can also graph the equation and see if it intersects the x-axis at any point, which would indicate a real root.
Yes, a quartic equation can have more than one real root. In fact, it can have up to four real roots, as the degree of the equation is 4. However, it is also possible for a quartic equation to have less than four real roots or no real roots at all.