Completing the Square

Completing the Square

Understanding the Principle

• Completing the Square is an algebraic technique used to solve quadratic equations, rearrange expressions, and find the roots of a function or equation.
• To complete the square of a quadratic (ax² + bx + c), the standard form is transformed to (a(x-h)² + k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.
• The process involves making the quadratic expression a perfect square trinomial.

Process Steps

• Begin by dealing with the coefficient of the x² term. If it’s not 1, it can be made 1 by factoring it out of the quadratic and linear terms.
• The second step involves writing the quadratic and linear terms as a square of a binomial.
• Find h by taking the coefficient of x (once the coefficient of x² is 1), dividing it by 2, and using the opposite sign. Write the expression as (x - h)².
• Finally, adjust the constant term to complete the square.

Applications

• Completing the square is crucial in several areas of mathematics. It’s used in the derivation of the quadratic formula.
• The Vertex Form of a quadratic function, which provides the vertex of the function and whether it opens upwards or downwards, can be easily written using this method.
• It’s used extensively in calculus, specifically in integration tasks, optimisation problems, and derivation of various mathematical theorems.

Practice and Mastery

• Get familiar with the process and the mathematical manoeuvres involved. Practice concern questions involving different problem types.
• Daily practise will help remember the process fluently.
• Always check the solution by substituting it back into the original equation.

Remember, mastering Completing the Square is a solid step to achieving proficiency in handling quadratic equations and functions.