Question

How do you solve `5s^{2} - 26s = 24`?

Answer 1

`x = -4/5 " or " x = -6`

Explanation:

Make a quadratic equation equal to 0

`5s^2 -26s-24 =0`

Now factorise:

Find factors of 5 and 24 which SUBTRACT (because of the MINUS 24) to give 26.

The signs will be DIFFERENT (because of the MINUS 24), there must be MORE negatives (because of the MINUS 26s)

Although this involves a bit of trial and error, you will get better with practice.

Use different combinations of factors, cross-multiply and subtract .

`color(white)(xx) color(red)5color(white)(xxx) color(blue)(4 rarr 1xx4 = 4)`
`color(white)(xx) color(blue)(1)color(white)(xxx) color(red)(6 rarr 5xx6 =ul30)`
`color(white)(xxxxxxxxxxxxxxx)26`

`(5x+4)(x-6) = 0`

Either bracket can be 0
`5x +4 = 0 rarr x = -4/5`

`x-6 = 0 rarr x = 6`

Answer 2

`-4/5 and 6`

Explanation:

Another way is solving it by the new Transforming Method (Socratic Search)
`y = 5s^2 - 26x - 24 = 0`
Transformed equation `y' = s^2 - 26x - 120 = 0`
Method: Find 2 real roots of y' then divide them by a = 5
Roots have opposite signs (ac < 0)
Compose factor pairs of (ac = - 120) --> (-3, 40)(-4, 30). This last sum is (30 - 4 = 26 = -b). Then, the 2 real roots of y' are: -4 and 30.
Back to y, its 2 real roots are: `s1 = -4/a = -4/5`, and `s2 = 30/a = 30/5 = 6`.

NOTE : This method avoids the lengthy factoring by grouping and solving the 2 binomials.