Completing The Square Of A Quadratic Trinomial

How to complete the square of a quadratic trinomial Quadratic trinomials are the polynomials with three terms and degree two.
Students in higher grades (grades 10 - 12) should know how to complete the square of a quadratic trinomial.
The standard form of a quadratic trinomial is given below: Ax² + Bx + C = 0 Where A is the coefficient of quadratic term (which is x²), B is the coefficient of the linear term (which is x) and letter C denotes the constant term.
To complete the square, the coefficient of the linear term "B" is divided by "2A" and then the result is squared, the obtained applying the above operations is used to complete the square.

Let's understand it by using the concrete example shown below: Complete the square of "y² + 8y + 6" The given trinomial have A = 1, B = 8 and C = 6.

To complete the square, add and subtract the following number: Number = (8/2)² = 16 Hence, add and subtract 16 to the given trinomial as shown below: y² + 8y + 16 - 16 + 6 Remember, 16 is added and subtracted to keep the same value of the expression. Now, y² + 8y + 16 is the perfect square and can be written as (y + 4)² and "- 16 + 6" can be solved to get "- 10".

As a result the given trinomial can be rewritten using the completed square term as shown below: = (y + 4)² - 10 Now if we solve "(y + 4)² - 10" by opening the brackets, we will get the original trinomial back.

Some times the coefficient of quadratic term is a number other than one as in the next example; 3x² - 12x + 6 First of all take the greatest common factor "3" out as shown in the next step: = 3 (x² - 6x + 2) Now complete the square taking similar steps to the previous example.

Add and subtract (6/2)² = 9 to the expression as in the next step; = 3 (x² - 6x + 9 - 9 + 2) Now, x² - 6x + 9 is a perfect square for (x - 3)² and the rest of the terms "- 9 + 2" can be solved to get "- 7".

After these steps we can rewrite the expression as in the next step: = 3 {(x - 3)² - 7} Next open the curl bracket to take "3" in as shown: = 3 (x - 3)² - 21 Above is the completed square for the second example.
Finally, the key to complete the square is 1.

Make the coefficient of quadratic term equal to one.
2.

Then divide the coefficient of linear term by two and square the result. 3.

Add and subtract the number obtained after the above operations to the given trinomial. 4.
First two term with the added constant completes the square.