Solve a simple equation using an assumption that the original has a solution. Construct an argument to justify this method. Also, resolve simple rational and radical equations in one variable and reflect on the causes of extraneous solutions. This includes solving linear equations and inequalities even when the coefficients are represented by letters.

Use completion of the square to convert any quadratic equation to the form (x – p)2 = q. Derive the quadratic formula from this form. Solve quadratic equations by various methods like taking square roots and factoring, and comprehend when the solutions are complex.

With a system of two equations in two variables, authenticate that substituting one equation with the sum of that equation and a multiple of the other gives an identical solution set. Solve linear systems of equations using accurate and approximate methods, and apply these to pairs of equations in two variables. Also, solve a linear and quadratic system in two variables using algebraic and graphic approaches, e.g., determining the intersection between a line and a circle.

Represent a linear system as a single matrix equation. Find a matrix inverse, if one exists, and use it to solve linear systems, employing technology for matrices bigger than 3 × 3. Understand that the graph