Solve each equation. 4x⁴+3x²-1 = 0

Key Concepts

Polynomial Equations

A polynomial equation is an equation that involves a polynomial expression, which is a sum of terms consisting of variables raised to non-negative integer powers and multiplied by coefficients. In this case, the equation 4x⁴ + 3x² - 1 = 0 is a fourth-degree polynomial equation, indicating that the highest power of the variable x is four. Understanding the structure of polynomial equations is essential for solving them.

Factoring and Roots

Factoring is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. Finding the roots of the polynomial, or the values of x that satisfy the equation, often involves factoring the polynomial or using the quadratic formula for lower-degree polynomials. In this case, recognizing that the equation can be transformed or factored is crucial for finding its solutions.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. When solving polynomial equations, especially those of degree four or higher, it is possible to encounter complex roots. Understanding how to work with complex numbers is important for fully solving equations that do not yield real solutions.