Question

How do you solve `x^2 - 12x + 80 = 8`?

Answer 1

`x= 6+-6i`
`color(white)"XXXX"`(there are no Real solution values)

Explanation:

Given `x^2-12x+80 = 8`

Subtracting `8` from both sides
`color(white)"XXXX"` `x^2-12x+72=0`

Using the quadratic formula (see below if you are uncertain of this)
`color(white)"XXXX"` `x= (-(-12)+-sqrt((-12)^2-4(1)(72)))/(2(1)`

`color(white)"XXXXXXXX"` `=(12 +-sqrt(144 - 288)/2`

`color(white)"XXXXXXXX"` `= (12+-sqrt(-144))/2`

`color(white)"XXXXXXXX"` `= (12+-12i)/2`

`color(white)"XXXXXXXX"` `=6+-6i`

Quadratic formula
Given the general quadratic equation in the form:
`color(white)"XXXX"` `ax^2+bx+c=0`
the solutions are given by the quadratic formula:
`color(white)"XXXX"` `x = (-b+-sqrt(b^2-4ac))/(2a)`

`color(white)"XXXXXX"`The need to use this formula comes up often enough that it is worth memorizing

Answer 2

`color(blue)(x=6(1+i)`
`color(blue)( x=6(1-i)`

Explanation:

`x^2−12x+80=8`
`x^2−12x+80 -8=0`

`x^2−12x+72=0` .

The equation is of the form `color(blue)(ax^2+bx+c=0` where:
`a=1, b=-12, c=72`

The Discriminant is given by:
`Delta=b^2-4*a*c`

`= (-12)^2-(4*1*72)`
`= 144-288=-144`

The solutions are found using the formula
`x=(-b+-sqrtDelta)/(2*a)`

`x = (-(-12)+-sqrt(-144))/(2*1) = (12+-sqrt(-144))/2`

`x=(12+-12i)/2`

`= (2(6+-6i))/2`

`= 6+-6i`

`color(blue)(x=6(1+i)`
`color(blue)( x=6(1-i)`