Solve each equation. 2x⁴-7x²+5=0

Verified step by step guidance

1. Recognize that the equation \$2x^{4} - 7x^{2} + 5 = 0\( is a quartic equation but can be treated as a quadratic in terms of \)x^{2}\(. To do this, let \)u = x^{2}$.
2. Rewrite the equation in terms of \(u\): \$2u^{2} - 7u + 5 = 0$.
3. Solve the quadratic equation \$2u^{2} - 7u + 5 = 0\( using the quadratic formula: \(u = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \)a=2\(, \)b=-7\(, and \)c=5$.
4. After finding the values of \(u\), substitute back \(u = x^{2}\) to get equations of the form \(x^{2} = u\).
5. Solve each equation \(x^{2} = u\) by taking the square root of both sides, remembering to consider both the positive and negative roots: \(x = \pm \sqrt{u}\).

Key Concepts

Polynomial Equations

Polynomial equations involve expressions with variables raised to whole-number exponents and coefficients. Understanding how to identify the degree and form of the polynomial is essential for choosing appropriate solving methods.

Substitution Method for Quadratic in Form

When a polynomial equation can be rewritten as a quadratic in terms of a new variable (e.g., letting y = x²), substitution simplifies solving higher-degree polynomials by reducing them to quadratic equations.

Solving Quadratic Equations

Quadratic equations can be solved using factoring, completing the square, or the quadratic formula. Mastery of these techniques is crucial once the polynomial is transformed into a quadratic form.