Question

How do you solve `2x^2+9x+10=0`?

Answer 1

`-5/2 and -2`

Explanation:

Use improved quadratic formula (Socratic Search)
`y = 2x^2 + 9x + 10 = 0`
Both roots are negative. (rule of signs)
`D = d^2 = b^2 - 4ac = 81 - 80 = 1` --> `d = +- 1`
There are 2 real roots:
`x = -b/(2a) +- d/(2a) = - 9/4 +- 1/4 = (-9 +- 1)/4`
`x1 = -10/4 = -5/2` and `x2 = -8/4 = -2`

Answer 2

`x = -2" "` or `" "x = -5/2`

Explanation:

Given:

`2x^2+9x+10 = 0`

We can use an AC method.

Find a pair of factors of `AC=2*10=20` with sum `B=9`

The pair `5, 4` works.

Use this pair to split the middle term and factor by grouping:

`0 = 2x^2+9x+10`

`color(white)(0) = 2x^2+5x+4x+10`

`color(white)(0) = (2x^2+5x)+(4x+10)`

`color(white)(0) = x(2x+5)+2(2x+5)`

`color(white)(0) = (x+2)(2x+5)`

Hence solutions:

`x = -2" "` or `" "x = -5/2`