Question

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

Answer 1

`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`