Question

How many solutions does `-12x^2-4x+5=0` have?

Answer 1

Two

Explanation:

It can only have 2 or less solutions because the highst power of x is 2 (`-12x^color(blue)(2)`). Lets check if it has 2, 1 or no solutions:

`-12x^2-4x+5=0|:(-12)`
`x^2+1/3x-5/12=0`
`color(blue)(x^2+1/3x+1/36)color(red)(-1/36-5/12)=0`
`color(blue)((x+1/6)^2)color(red)(-16/36)=0|+16/36`
`(x+1/6)^2=16/36|sqrt()`
`x+1/6=+-2/3|-1/6`
`x=+-2/3-1/6`
`x_1=1/2 or x_2=-5/6`

Answer 2

Method shown below, you do the math.

Explanation:

Rewrite the equation, change signs on both side:

`12x^2 +4x -5 =0`
This can be seen to be the familar quadratic equation

`ax^2 +bx +c` with a solution:

`x= ( -b +- sqrt(b^2 -4ac))/(2a)`

Substitute values to a, b, c to get the answer