Completing the square
The idea of completing the square is to add something to an equation to make that equation a perfect square. This makes solving a lot of equations easy. In fact, all quadratic equations can be solved by completing the square.
As an example, we have the equation . Look at the part. If was added to this, then we would have a perfect square, . To do this, add to each side of the equation to get
All quadratic equations in the form can be solved by taking the square root of and subtracting . Completing the square is a technique to manipulate every quadratic into the easily solvable form above.
General Solution For A Quadratic by Completing the Square
Let the quadratic be in the form , with .
Subtracting from both sides of the equation, we obtain
Dividing by and adding to both sides yields
Factoring the LHS gives
As described above, an equation in this form can be solved, yielding
This formula is also called the quadratic formula.
Applications of Adding and Factoring
Other degrees of polynomials may be solved by adding constant terms and factoring.
Another common usage is in conic sections. The equations for conic sections typically contain a squared term such as . However, the problem may be posed as to convert from an expanded form to a factored perfect square. Completing the square is the standard method.
All kinds of exotic factoring techniques are used on the AIME, including completing the square.
solve for x
What is the value of if