Step-by-step explanation:
you did not tell us what we need to create or find or calculate.
I guess from the picture this is one of the new methods to factoring equations. typically where the method is often more complicated than just thinking logically and doing it.
let's see if I can figure that one out.
but first, the basic, normal approach.
factoring the equation means that find in this case (as it is a quadratic equation) 2 terms in x in the form
(ax + b)(cx + d) = 10x² - x - 2
with a, b, c, d being integer values.
after doing the multiplications we get
acx² + bcx + adx + bd = 10x² - x - 2
acx² + (bc + ad)x + bd = 10x² - x - 2
we see
ac = 10
bc + ad = -1
bd = -2
10 can be factored as
10×1
-10×-1
5×2
-5×-2
-2 can be factored as
-1×2
1×-2
so, get -1 for the middle one, I am trying to get -5 + 4. and that is possible by saying
b = -1
c = 5
d = 2
a = 2
to check
ac = 2×5 = 10
bc + ad = -1×5 + 2×2 = -5 + 4 = -1
bd = -1×2 = -2
that works perfectly.
so the factoring is
(2x - 1)(5x + 2) = 10x - x - 2
now to the box method :
the general quadratic equation is
ax² + bx + c
that means for us
a = 10
b = -1
c = -2
the box square starts with
-2
10x²
now we find 2 numbers that multiply across to a×c = -20 and add together to b = -1
n×m = -20
n+m = -1
n = -1 - m
and that we use in the first equation
(-1 - m)×m = -20
-m² - m + 20 = 0
m² + m - 20 = 0
using the general solution for such quadratic equations
m = (-1 ± sqrt(1² - 4×1×-20))/(2×1) = (-1 ± sqrt(81))/2 =
= (-1 ± 9)/2
m1 = (-1 + 9)/2 = 8/2 = 4
m2 = (-1 - 9)/2 = -10/2 = -5
as we see (n×m = -20) m1 = n2 and m2 = n1
so, the 2 numbers (sequence irrelevant) are -5, 4.
they represent -5x and 4x. the negative one goes in the upper enough box.
-5x -2
10x² 4x
now we create a column to the left of the box table, and a row below the box table.
the left column gets the greatest common factor of the line elements.
the greatest common factor of -5x and -2 is -1.
the greatest common factor of 10x² and 4x is 2x.
-1 -5x -2
2x 10x² 4x
the bottom row divides the last row by the left column element
10x² / 2x = 5x
4x / 2x = 2
-1 -5x -2
2x 10x² 4x
5x 2
and the factorization is then the left column and the bottom row :
(2x - 1)(5x + 2)
now anybody tell me this is not much more complicated than the direct method.
this is only advisable, if the direct method has too many options to try and combine.
I hope I found the right thing your teacher wants here.
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