Completing the square is an algebraic technique used to solve quadratic equations and rewrite them in a more convenient form. This method involves transforming the quadratic equation from the standard form: into the form:
In doing so, we are effectively ‘completing’ the square of the quadratic expression by adding and subtracting a specific value to make it a perfect square trinomial.
A perfect square trinomial is a quadratic expression that can be factored into the product of two identical binomials. For example:
This quadratic expression can be factored into the product of two identical binomials:
Simplifying this expression, we get:
So, is a perfect square trinomial that can be factored into .
Completing the square is very helpful when factoring or using the quadratic formula isn’t easily applicable. It can also be used to find the highest or lowest point of a curve represented by the quadratic equation.
Before we look at a worked example, let’s look through the steps for completing the square:
Step 1: Write the quadratic equation in standard form .
Step 2: If the coefficient of x^2 (a) is not 1, factor it out from the quadratic terms.
Step 3: Divide the coefficient of x (b) by 2, square the result, and add and subtract it within parentheses.
Step 4: Combine the terms in the parentheses to form a binomial squared and rewrite the remaining constant outside the parentheses as e.
Now, let’s look at a worked example:
Solve the quadratic equation
Our goal is to rewrite this equation in the form of (x – d)^2 + e = 0, where d and e are constants. To achieve this, we’ll complete the square. Here are the steps:
1. Identify the coefficient of the linear term, which is -6 in this case.
2. Take half of this coefficient:
3. Square the result from step 2:
4. Add and subtract the result from step 3 to the left side of the equation:
5. Factor the first three terms into a perfect square:
Now, the equation is in the form of , where d = 3 and e = –4.
To solve for x, you can either use the square root property or the quadratic formula.
Using the square root property:
So, the solutions are
Therefore, the solutions for the quadratic equation are and .
The process is slightly different when the coefficient of the term is not 1. This is because you have to factor the coefficient a of the term out of the quadratic and linear terms, leaving you with .
Solve the equation
Solve the equation
Write the expression in the form , where a, b and c are constants.