Solve each equation. 4x⁴+3x²-1 = 0

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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 \(y = x^{2}\).

Rewrite the equation in terms of \(y\): \(4y^{2} + 3y - 1 = 0\).

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 \(c=-1\).

After finding the values of \(y\), substitute back \(y = x^{2}\) and solve for \(x\) by taking the square root: \(x = \pm \sqrt{y}\).

Check each solution for \(x\) to ensure it satisfies the original equation, and write the final solution set.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Equations

Polynomial equations involve expressions with variables raised to whole-number exponents and coefficients. Solving them means finding all values of the variable that satisfy the equation. Understanding the degree and structure of the polynomial helps determine the methods used for solving.

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 a lower power term (e.g., x⁴ as (x²)²), substitution simplifies it into a quadratic equation. This method involves letting a new variable represent the lower power term to solve more easily.

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 substitution reduces the original equation to quadratic form, these techniques help find the roots, which can then be back-substituted to find the original variable values.

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