Completing the square

Completing the Square

Introduction

  • Completing the square is an important algebraic method used to simplify quadratic equations, making them easier to solve.
  • This method transforms quadratics from the standard form, ax^2 + bx + c, into the vertex form, a(x - h)^2 + k.

Procedure for Completing the Square

  • Identify the coefficients “a”, “b”, and “c” in the quadratic equation.
  • If “a” is not equal to 1, divide every term by “a” to make the coefficient 1.
  • Move the constant term “c” to the other side of the equation by using subtraction.
  • Rewrite the equation and add (b/2)^2 on both sides of the equation.
  • Factorise the left side of the expression to get (x + b/2)^2.
  • Simplify the right side of the equation to find “k”.

Applying Completing the Square

  • Completing the square is essential when solving quadratic equations, particularly when they cannot be easily factored.
  • This technique is also used in the derivation of the quadratic formula and in sketching the graph of a quadratic function.
  • It can also be applied to solve equations involving circles and ellipses in coordinate geometry.

Examples

  • For instance, assume you have x^2 + 6x + 5 = 0. First, consider only x^2 + 6x and take the coefficient of x, which is 6, divide by 2 and square it to get 9. Add 9 to both sides of the equation, leading to x^2 + 6x + 9 = 9 + 5. This can be rewritten as (x + 3)^2 = 14, which represents the completed square.
  • For an example with a non-1 leading coefficient, consider 2x^2 + 4x - 3 = 0. First, divide through by “2” to have x^2 + 2x - 3/2 = 0, then move the constant to the right hand side to have x^2 + 2x = 3/2. Add (1)^2 to both sides to complete the square, to obtain x^2 + 2x + 1 = 3/2 + 1. Factorise the left side to get (x + 1)^2 = 5/2.

Conclusion

  • Mastering completing the square is essential to performing well in algebra.
  • Regular practice will increase speed and accuracy, and enhance your mathematical problem-solving skills.
  • Always start by identifying “a”, “b”, and “c”, and remember to add (b/2)^2 to both sides of the equation to achieve the completed square.