Question

How do you solve using the completing the square method `2x^2-9x-17=0`?

Answer 1

`x = 9/4+-sqrt(217)/4`

Explanation:

In addition to completing the square, I will use the difference of squares identity:

`a^2-b^2 = (a-b)(a+b)`

with `a = (4x-9)` and `b = sqrt(217)`.

To cut down on the need for fractions, multiply through by `2^3 = 8` first to get:

`0 = 16x^2-72x-136`

`=(4x-9)^2-81-136`

`=(4x-9)^2-217`

`=(4x-9)^2-(sqrt(217))^2`

`=((4x-9)-sqrt(217))((4x-9)+sqrt(217))`

`=(4x-9-sqrt(217))(4x-9+sqrt(217))`

`=16(x-(9+sqrt(217))/4)(x-(9-sqrt(217))/4)`

So `x = 9/4+-sqrt(217)/4`