Solving a Bi-Quadratic Equation

Basic Concept

• Objective: Solving a bi-quadratic equation (degree 4)
• Approach:
  • Reduce degree from 4 to cubic (degree 3) and then to quadratic (degree 2)
  • Solve the quadratic equation using familiar methods like the quadratic formula or term splitting

Key Steps

Setting Up Another Bi-Quadratic Equation

• Consider an equation of the form:
  • ( X^2 + \frac{1}{2} \text{(coefficient of } X) \cdot X + \lambda )^2
  • This can be expanded using ( (a + b + c)^2 )
  • Collect all relevant like terms and simplify

Transforming the Equation

• Move all terms except the constructed bi-quadratic to the right-hand side
• Collect like terms together:
  • Highest power terms: (X^2)
  • Coefficient of (X)
  • Constant terms
• Ensure the right-hand side equals a perfect square

Discriminant Concept

• If the right side is a perfect square, you have equal and real roots
• Set the discriminant (D) to zero ((b^2 - 4ac = 0))

Solving for Lambda

• Substitute values and solve for (\lambda)
• Expand and simplify
• Check for roots via factorization or trial and error (e.g., checking for (\lambda = -1))

Obtaining Roots

1. Factorize the cubic equation obtained: Identify one root and reduce to quadratic form

• Example: If (\lambda = -1) is a root, substitute back into the equation

2. Form quadratic equations from the reduced form

• Simplify quadratic equations to obtain roots

3. Applying quadratic formula or term splitting

• Ensure you obtain all four roots as required for a bi-quadratic equation

Final Roots

• Solve each quadratic equation to find all four roots
• Ensure you cover all possibilities

Summary

• Goal: Simplify bi-quadratic to quadratic forms and solve
• Methods used: Transformations, factoring, quadratic formula
• Result: Four roots for the bi-quadratic equation

Example

• Given equation: (X^2 - X + \lambda)^2 simplified forms and steps shown leading to roots
• Sample roots: (\frac{3 \pm \sqrt{5}}{2}) and (-3, 1)

End of Lecture

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Takeaway

• Understanding and solving bi-quadratic equations step-by-step using basic algebraic techniques.