Solve by completing the square

Introduction

The process of solving equations by completing the square is a strategy used to solve quadratic equations and find the roots.
It essentially involves transforming a quadratic equation into the form (x-h)^2 = k, from which the solutions can be easily found.

Process of Completing the Square

Start by ensuring that the coefficient of the x^2 term is 1. If it’s not, divide all terms by the coefficient of the x^2 term.
The next step is to rearrange the equation such that the x^2 term and the x term are on one side, and the constant is on the other. For example, if you have x^2 + 4x - 3, rearrange it to x^2 + 4x = 3.
The main step in completing the square involves adding the square of half the coefficient of x to both sides of the equation. For example, if the coefficient of x is 4, half of it is 2, and square of 2 is 4. So add 4 to both sides of the equation from the previous step to rewrite it as x^2 + 4x + 4 = 3 + 4.
The left-hand side of the equation will now be a perfect square, and this can be written in shortened form. In the previous example, the equation x^2 + 4x + 4 (which is equal to (x + 2)^2) becomes (x+2)^2 = 7.

Solving for x

Once you have an expression in the form (x - h)^2 = k, you can easily solve for x by taking the square root of both sides.
Remember to take both the positive and negative square roots. So for the equation (x+2)^2 = 7, the solutions would be x = -2 + √7 and x = -2 - √7.

Key Points

The main advantage of solving by completing the square is its application to problems where the quadratic doesn’t factorise easily.
This process not only helps in solving equations, but also aids in sketching graphs of quadratic equations as the form (x - h)^2 +k immediately gives the vertex of the parabola.
Mastering this technique requires practice, so work through many different examples until the process feels natural. Always check your solutions either by substituting back into the original equation or using another method.