Applications of completing the square

Applications of completing the square

Introduction to Completing the Square

  • Completing the square is a method used to solve quadratic equations, simplify expressions and derive a graph of a quadratic function.
  • It is named for its use in converting a standard form quadratic to the form of (x-h)² + k, which resembles a perfect square trinomial.

Principles of Completing the Square

  • In completing the square, the quadratic equation is re-arranged into a form that can be easily solved.
  • This method is particularly useful if the quadratic equation does not factorise simply, or at all.
  • The goal is to create a ‘square’ on one side of the equation, hence the name.

Process of Completing the Square

  1. The first step is to ensure that the coefficient in front of the x² term is ‘1’. If not, you should divide each term in the equation by that coefficient.
  2. Next, to create a perfect square trinomial, add the square of half the x term to both sides of the equation.
  3. Recognise the left-hand side as a perfect square and simplify the right-hand side.
  4. Write the equation as (x-h)² = k, a standard form of a quadratic equation.

Examples

  • For instance, to complete the square for x² + 4x - 5 = 0, you add the square of half the x term (4/2)² = 4 ensuring to keep the equation balanced.
  • Resultantly, you get x² + 4x + 4 - 5 - 4 = 0, which simplifies to (x + 2)² - 9 = 0.

Applications of Completing the Square

  • Completing the square has vital real world applications like calculating the area and volume of physical objects, understanding the motion of objects, and solving optimization problems in industrial processes.
  • It is also used in geometry for deriving the equation of circles and understanding the properties of parabolas. For example, the equation of a circle is usually given in the completed square form.
  • Completing the square is used in chemistry too, in calculating reaction rates and in the study of molecular structures.

Extra Notes

  • Always check that the coefficient of the x² term is ‘1’ when starting to complete the square.
  • Remember to balance the equation by adding or subtracting the same value from both sides.
  • Being able to complete the square comes in handy, especially when solving quadratic equations that do not factorise simply.