Question

How do you solve `x^2-2x-12=0` algebraically?

Answer 1

`x~~4.606" " -2.606` to 3 decimal places

`color(green)(x=1+-sqrt(13))`

Explanation:

The whole number factors of 12 are {6,2}, {3,4} and none of these have a difference of 2 ( from `2x`). So it must have none integer solutions for `x`. Thus we need to use the formulas. These formula are something worth remembering.

It takes quite a bit of work to make them stick!

Standard equation form:`" "y=ax^2 +bx+c`

where `color(magenta)(x=(-b+-sqrt(b^2-4ac))/(2a))` ............................(1)

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Another one worth remembering is the completed square format.
Otherwise known as Vertex Form.

`" "color(magenta)(y=a(x+b/(2a))^2+c -[(b/2)^2])` .............................(2)

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`color(blue)("Using equation (1)")`

`=>x" "=" "(-(-2)+-sqrt( (-2)^2-4(1)(-12)))/(2(1))`

`=>x" "=" "(2+-sqrt( 52))/2`

`=>x" "=" "(2+-sqrt(2^2xx13))/2`

`color(green)(=>x= 1+-sqrt(13))`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Using equation (2)")`

`y=1(x-1)^2-12-1`

`y=(x-1)^2-13`

Set `y=0`

`0=(x-1)^2-13`

Add 13 to both sides

`13=(x-1)^2`

Square root both sides

`+-sqrt(13)=x-1`

Add 1 to both sides

`color(green)(x=1+-sqrt(13))`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`x~~4.606" " -2.606` to 3 decimal places