Solve each equation. 4x⁴+3x²-1 = 0

Verified step by step guidance

1. Recognize that the equation \$4x^{4} + 3x^{2} - 1 = 0\( is a quartic equation but can be treated as a quadratic in terms of \)x^{2}\(. To do this, let \)y = x^{2}\(, which transforms the equation into \)4y^{2} + 3y - 1 = 0$.
2. Solve the quadratic equation \$4y^{2} + 3y - 1 = 0\( using the quadratic formula: \)y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\(, where \)a=4\(, \)b=3\(, and \)c=-1$.
3. Calculate the discriminant \(\Delta = b^{2} - 4ac = 3^{2} - 4 \cdot 4 \cdot (-1)\) to determine the nature of the roots for \(y\).
4. Find the two values of \(y\) by substituting the discriminant back into the quadratic formula. These values correspond to \(x^{2}\).
5. For each value of \(y\), solve for \(x\) by taking the square root: \(x = \pm \sqrt{y}\). Remember to consider both the positive and negative roots for each \(y\).

Key Concepts

Polynomial Equations

Polynomial equations involve expressions with variables raised to whole-number exponents and coefficients. Solving these equations means finding all values of the variable that make the equation true. Understanding the degree of the polynomial helps determine the number of possible solutions.

Substitution Method for Quadratic in Form Equations

When a polynomial equation can be rewritten so that a higher power term is expressed as a square of another variable (e.g., x⁴ as (x²)²), substitution simplifies it into a quadratic equation. This method involves letting a new variable represent the inner power, solving the quadratic, then back-substituting to find the original variable's values.

Solving Quadratic Equations

Quadratic equations are second-degree polynomials that can be solved by factoring, completing the square, or using the quadratic formula. Once the equation is in standard form, these methods help find the roots, which can be real or complex numbers, depending on the discriminant.