Mathematics Foundations/8.4 Solving Equations

8.4 Solving Equations Introduction This section explores methods for finding solutions to various types of equations. While the Fundamental Theorem of Algebra guarantees the existence of solutions for polynomial equations, finding these solutions explicitly often requires specialized techniques. Polynomial Equations Linear Equations Definition 8.4.1: A linear equation has the form where . The solution is given by . Quadratic Equations Definition 8.4.2: A quadratic equation has the form where . The solutions are given by the quadratic formula: The discriminant determines the nature of the roots: If , there are two distinct real roots If , there is one repeated real root If , there are two complex conjugate roots Cubic Equations Definition 8.4.3: A cubic equation has the form where . Cardano's Method: Substitute to eliminate the term, obtaining Set and solve for and using the constraint This leads to and being the roots of Solve for and , then find Substitute back to find Quartic Equations Definition 8.4.4: A quartic equation has the form where . Ferrari's Method: Transform to by dividing by Substitute to eliminate the term Rearrange to Add to both sides for some parameter Factor the left side as a perfect square Choose to make the right side a perfect square This leads to a cubic equation (the "resolvent") Solve the resolvent to find Use to factor the quartic and solve the resulting quadratics Higher-Degree Equations Theorem 8.4.5 (Abel-Ruffini): There is no general algebraic formula (using only addition, subtraction, multiplication, division, and radicals) for solving polynomial equations of degree five or higher. For specific types of higher-degree equations, specialized techniques exist: Reciprocal equations Equations with special symmetries Equations with known factors Transcendental Equations Exponential Equations Equations involving exponential functions, such as or . Example 8.4.6: To solve : Logarithmic Equations Equations involving logarithmic functions, such as . Example 8.4.7: To solve : Trigonometric Equations Equations involving trigonometric functions. Example 8.4.8: To solve for : or Systems of Equations Linear Systems Systems of linear equations can be solved using: Substitution method Elimination method Matrix methods (Gaussian elimination, Cramer's rule) Example 8.4.9: Consider the system: Using elimination: Multiply equation (2) by 3: Add equations (1) and (3): Substitute back into equation (1): Nonlinear Systems Systems involving nonlinear equations can be solved using: Substitution Elimination Transformation techniques Numerical methods Numerical Methods Bisection Method Algorithm 8.4.10: For a continuous function with : Set If , return as the solution If , set ; otherwise, set Repeat until desired accuracy is achieved Newton-Raphson Method Algorithm 8.4.11: For a differentiable function : Start with an initial guess Compute Repeat until for some small Fixed-Point Iteration Algorithm 8.4.12: Rewrite as , then: Start with an initial guess Compute Repeat until Exercises Exercise 8.4.1: Solve the cubic equation given that one root is . Exercise 8.4.2: Find all solutions to in the interval . Exercise 8.4.3: Solve the system: Exercise 8.4.4: Use the Newton-Raphson method to find an approximate solution to with starting value . Perform three iterations. Exercise 8.4.5: Prove that the equation has exactly one real root.