Question

How do you solve quadratic equation `4x^2+11x-20=0`?

Answer 1

`x = 5/4 and x = 4`

Explanation:

The first method to check for solving a quadratic equation is whether the expression factorises.

`4x^2 +11x-20 = 0`

"Find factors of 4 and 20 which subtract to make 11"

Note that 11 is ODD, so the factors must combine to give one ODD and and one even number.

That immediately eliminates `2xx2` and `10xx2` as possible factors of 4 and 20
( because their multiples will always be even.)

When trying different combinations, remember not to have a common factor in any horizontal row.

Find factors and cross-multiply. Subtract the products to get 11.

`" "ul(4" "20)`
`" "4" "5" "rarr1 xx 5 = 5`
`" "1" "4" "rarr 4xx4 = ul16`
`color(white)(xxxxxxxxxxxxxxxxxxx)11` the difference is 11

We have the correct factors, now work with the signs.

MINUS `20` means that the signs must be different.
PLUS `11` means there must be more positives.
Fill in the correct signs, starting from `color(red)(+11)`

`color(red)(+11) " "rarr color(red)(+16) and color(blue)(-5)`

`" "ul(4" "20)`
`" "4" "5" "rarr 1 xx 5 = color(blue)(-5)`
`" "1" "4" "rarr 4xx4 = ulcolor(red)(+16)`
`color(white)(xxxxxxxxxxxxxxxxxxx)color(red)(+11)`

Now fill in the signs next to the correct factors:

`" "ul(4" "20)`
`" "4" "color(blue)(-5)" "rarr 1 xx color(blue)(-5) = color(blue)(-5)`
`" "1" "color(red)(+4)" "rarr 4xxcolor(red)(+4) = ulcolor(red)(+16)`
`color(white)(xxxxxxxxxxxxxxxxxxxxx)color(red)(+11)`

Now you have the factors: Top row is one bracket and bottom row is the other factor.

`4x^2 +11x-20 = 0`
`(4x-5)(x+4) =0`

Letting each factor be equal to 0 gives the 2 solutions

`4x-5 =0 " "rarr " " 4x=5 rarr x= 5/4`
`x+4=0" "rarr" "x=-4`

Answer 2

`5/4 and -4`

Explanation:

`y = 4x^2 + 11x - 20 = 0`.
Use the new Transforming Method (Socratic Search)
Transformed equation: `y' = x^2 + 11x - 80 = 0`.
First, find the 2 real roots of y' that have opposite signs (ac < 0). Then, divide them by (a).
Factor pairs of (- 80) --> ...(4, - 20)(5, -16). This last sum is (-11 = -b). There for the 2 real roots of y' are: 5 and - 16.
The 2 real roots of y are: `5/4` and `- 16/4 = - 4`