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

2 Answers
Nov 26, 2016

Answer:

#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#

Jan 12, 2017

Answer:

#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#