Completing the Square – iGCSE Mathematics A Edexcel Revision

“Completing the Square” is a method used for solving quadratic equations, rewriting quadratic expressions, and graphing quadratic functions.
Quadratic equations take the form ax² + bx + c = 0. The term “completing the square” refers to the process of creating a perfect square trinomial from the quadratic equation.
To complete the square, the first step is to make sure that your quadratic equation is in the standard form a(x-h)² + k. You do this by transferring any constant term (c, in our standard form) over to the other side of the equation.
The next step in completing the square is to deal with ‘b’, the coefficient of the x term. We halve ‘b’ and then square the result. This number is then added and subtracted to the equation, essentially adding zero, so as not to change the value of the equation.
This process allows us to convert the quadratic function into the completed square form. This eventually forms a pattern with a square of a binomial on one side.
The primary reason for completing the square is to help solve the quadratic equation. The square root property states that if p² = q, then p = ± √q can be used to solve for ‘p’.
This method also aids in finding the axis of symmetry for a quadratic equation, a helpful tool in graphing. The axis of symmetry is given by the formula x = -b/(2a).
You should also understand how to use the vertex form of the equation f(x) = a(x-h)² + k to find the vertex of a parabola. The vertex of the parabola is given by the point (h, k).
Also, keep in mind that the “a” variable in the vertex form of a quadratic function impacts the width and direction of the parabola. If “a” is positive, the parabola opens upwards, and if “a” is negative, the parabola opens downwards.
Remember to practice several problems of different scenarios to fully understand the process and applications of completing the square. You can solve quadratic equations by factoring, using the quadratic formula, or completing the square. With practice, you’ll get to determine which method is most suitable for a given problem.
Lastly, take a look at the discriminant. The discriminant (b² - 4ac from the quadratic formula) determines the number of roots of a quadratic equation and can help you determine the best method of solving it.