Fill in the number that makes the polynomial a perfect-square quadratic. z2 + 18z +

Answer 1

When the leading coefficient is 1 (which is true in this case), just think 1/2 of b squared to get the constant term that will turn this into a perfect square trinomial.

For example:

x² - 10x + ____

Take 1/2 (10) = 5, then 5²=25. 25 "completes the square".

x² - 10x + 25 = (x - 5)²

If the coefficient of the linear term is not an even number you will start getting fractions:

x² + 9x + ____

Since the leading coefficient is 1, take half of 9 and square that result to complete the square:

(9/2)²= 81/4

So,

x² + 9x + 81/4 = (x + 9/2)²

When the leading coefficient is not 1, probably the easiest way to proceed is to factor out the leading coefficient, complete the square on the trinomial, then either redistribute the leading coefficient or leave it factored out.

So,

2x² + 16x + ___

Rewrite as:

2(x² + 8x + ___)

2(x² + 8x + 16)

2(x + 4)²

For your problem:

Leading coefficient is 1, so half of 18 = 9, and 9²=81 (81 completes the square).

x² +18x + 81

to rewrite as a binomial squared just take half of the 18

(x + 9)².

This technique comes in handy when solving quadratic equations:

x² + 6x = 25

x² + 6x + 9 = 25+9 (complete the square on the left, adding 9 to both sides of the equation)

(x + 3)²=34 (rewrite the perfect square trinomial as a binomial squared)

x+3 = ±√34 (square root property)

x= -3 ±√34

That is a lot easier than using the quadratic formula!