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Sagot :
EXPLANATION
Given the equation:
[tex]x^2\text{ + 7x -18 = 0}[/tex]We can apply the quadratic equations formula as shown as follows:
Break the expression into groups:
For:
[tex]ax^2+bx+c[/tex]Find u,v such that u*v = a*c and u+v = b and group into (ax^2+ux)+(vx+c)
a=1, b=7, c=-18
u*v=-18, u+v = 7
Find the primer factors of 18:
18 / 2 = 9
9 / 3 = 3
2,3 are all prime numbers, therefore no further factorization is possible.
Multiply the prime factors of 18: 6,9
Add the prime factors: 2,3
Add 1 and the number 18 itself
1, 18
The factors of 18:
1, 2, 3 , 6, 9, 18
Negative factors of 18:
Multiply the factors by -1 to get the negative factors:
-1, -2, -3, -6, -9, -18
For every two factors such that u*v=-18 , check if u+v = 7:
[tex]\mathrm{Check}\: u=1,\: v=-18\colon\quad \: u\cdot v=-18,\: u+v=-17\quad \Rightarrow\quad \mathrm{False}[/tex][tex]\mathrm{Check}\: u=2,\: v=-9\colon\quad \: u\cdot v=-18,\: u+v=-7\quad \Rightarrow\quad \mathrm{False}[/tex][tex]\mathrm{Check}\: u=3,\: v=-6\colon\quad \: u\cdot v=-18,\: u+v=-3\quad \Rightarrow\quad \mathrm{False}[/tex][tex]\mathrm{Check}\: u=6,\: v=-3\colon\quad \: u\cdot v=-18,\: u+v=3\quad \Rightarrow\quad \mathrm{False}[/tex][tex]\mathrm{Check}\: u=9,\: v=-2\colon\quad \: u\cdot v=-18,\: u+v=7\quad \Rightarrow\quad \mathrm{True}[/tex][tex]\mathrm{Check}\: u=18,\: v=-1\colon\quad \: u\cdot v=-18,\: u+v=17\quad \Rightarrow\quad \mathrm{False}[/tex]u=9, v=-2
Group into:
[tex](ax^2+ux)+(vx+c)[/tex][tex](x^2-2x)+(9x-18)[/tex]Factor out x from x^2 -2x
x^2 -2x
Factor out common term x:
=x(x-2)
Factor out 9 from 9x - 18:
Rewrite 18 as 2*9:
9x - 9*2
Factor out common term 9:
9(x-2)
=x(x-2) + 9(x-2)
Factor out common term x-2:
[tex]=\mleft(x-2\mright)\mleft(x+9\mright)[/tex]The solution to the quadratic equation x^2 + 7x - 18 = 0 applying the factorizing method is:
[tex]=(x-2)(x+9)[/tex]
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