How do you solve using the completing the square method # x^2-14x+45=0#?

1 Answer
May 2, 2016

Answer:

#x = 9# OR #x = 5#

There must be two answers because it is a quadratic equation.

Explanation:

Completing the square is a method based on the product of the square of a binomial: #(x + m)^2 = x^2 +2xm + m^2#
This is in the form of a quadratic equation : #ax^2 +-bx + c#

#(x + 3)^2 = x² + 6x + 9#
#(x - 6)^2 = x² - 12x + 36#

Notice that
#a = 1#
the first and last terms are always perfect squares.
There is a specific relationship between #b and c#

#(b÷2)^2# gives the value of c. (half of #b#, then square the answer.)

If you have a trinomial in this form it can be written as #(x +- ..)^2#

Complete the following square: #x^2 + 10x + .......#
Do you see that the missing value is 25?

#x^2 + 10x + 25# can be written as #(x+5)^2#

Let's look at your question:

In #x² - 14x + 45 = 0#, 45 is not the correct value for c.

Move 45 to the other side: #x² - 14x ....... = -45#

Add the required value TO BOTH SIDES
#x² - 14x + color(red)49 = -45 + color(red)49#

now: #(x - 7)^2 = 4# ................... [7 is half of 14 or# sqrt49#]

#x-7# = #+-2# .........................find the square root of both sides

#x = 2+7# OR #x = -2+7#

#x = 9# OR #x = 5#