= 6x² + 19x - 7
First method : discriminant
Polynomial like : ax² + bx + c, where :
a = 6
b = 19
c = - 7
Δ = b² - 4ac (discriminant)
Δ = 19² - 4(6 * - 7) = 361 + 168 = 529 = 23²
x1 = (- b - √Δ) / 2a = (- 19 - 23) / (2 * 6) = 42/12 = 7/2
x2 = (- b + √Δ) / 2a = (- 19 + 23) / (2 * 6) = 4/12 = 1/3
The polynomial can be written : a(x - x1)(x - x2)
= 6[x - (7/2)][x - (1/3)]
= 2[x - (7/2)] * 3[x - (1/3)]
= (2x - 7)(3x - 1)
Second method : grouping
I'm French, and I don't know this method, but I discovered it on Internet. it's interesting.
http://www.youtube.com/watch?v=XmuF1ACkgCA
Adapt to your case : 6x² + 19x - 7
This polynominal looks like : ax² + bx + c
First, you calculate : a * c = - 42
Then, find the factors of - 42, the sum of them must be equal to b, so : 19
42 = - 1 * 42 : sum = - 1 + 42 = 41 ≠ 19 wrong
42 = - 2 * 21 : sum = - 2 + 21 = 19 OK
Second, you write : 19 = - 2 + 21, and the polynomial becomes :
= 6x² + (- 2 + 21)x - 7, the expand with distributivity
= 6x² - 2x + 21x - 7
Third, try to factorize, it's so easy :
example 1 : 6x² - 2x = 2x(3x - 1)
example 2 : 21x - 7 = 7(3x - 1)
This polynominal becomes :
= 2x(3x - 1) + 7(3x - 1)
Fourth, fatorise a last time, it's evident : (3x - 1)
= (3x - 1)(2x + 7)
Therefore, you can write : 6x² + 19x - 7= (3x - 1)(2x + 7)
Set up possible factors of your a value, 6, and your c value, 7
factors of 6:
1, 6
2,3
Factors of -7:
1, -7
-1, 7
Play around with the correpsonding factors so that by multiplying them, they add up to the b value, 19
lets try 2, 3, and -1 7
so arrage them vertically to get:
3 -1
2 7
now cross multiply:
2*7 +(-1*2)=19. thus we know these are the factors of this polynomial,
therefore the answer is simply 6x^2 +19x -7 = (3x -1)(2x +7)
Hope this helps!
(7 + 2 x) (-1 + 3 x)
6x^2+19x-7 =
(2x + 7)(3x - 1)
You can't the only factors of -7 are 1 & and 7.
Also there is no common factor to pull out.
Hi,
Here is what I've learned in school:
Let's say you have the following equation: a*x^2+b*x+c=0 .
Calculate Delta = b^2 - 4*a*c
First solution: x1 = (-b + sqrt(Delta)) / (2*a)
Second solution: x2 = (-b - sqrt(Delta) / (2*a)
In your case: a=6, b=19, c=-7.
Delta = 529
x1 = 4/12 = 0.33
x2 = -42/12 = -3.5
The equation can be written as a*(x-x1)*(x-x2):
6*(x-4/12)*(x+42/12) = (3x-1)(2x+7)
6x^2 + 19x - 7
6x^2 + 21x - 2x - 7
(6x^2 + 21x) - (2x - 7)
3x(2x + 7) - 1(2x + 7)
(3x - 1)(2x + 7)
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Answers & Comments
Verified answer
= 6x² + 19x - 7
First method : discriminant
Polynomial like : ax² + bx + c, where :
a = 6
b = 19
c = - 7
Δ = b² - 4ac (discriminant)
Δ = 19² - 4(6 * - 7) = 361 + 168 = 529 = 23²
x1 = (- b - √Δ) / 2a = (- 19 - 23) / (2 * 6) = 42/12 = 7/2
x2 = (- b + √Δ) / 2a = (- 19 + 23) / (2 * 6) = 4/12 = 1/3
The polynomial can be written : a(x - x1)(x - x2)
= 6[x - (7/2)][x - (1/3)]
= 2[x - (7/2)] * 3[x - (1/3)]
= (2x - 7)(3x - 1)
Second method : grouping
I'm French, and I don't know this method, but I discovered it on Internet. it's interesting.
http://www.youtube.com/watch?v=XmuF1ACkgCA
Adapt to your case : 6x² + 19x - 7
a = 6
b = 19
c = - 7
This polynominal looks like : ax² + bx + c
First, you calculate : a * c = - 42
Then, find the factors of - 42, the sum of them must be equal to b, so : 19
42 = - 1 * 42 : sum = - 1 + 42 = 41 ≠ 19 wrong
42 = - 2 * 21 : sum = - 2 + 21 = 19 OK
Second, you write : 19 = - 2 + 21, and the polynomial becomes :
= 6x² + (- 2 + 21)x - 7, the expand with distributivity
= 6x² - 2x + 21x - 7
Third, try to factorize, it's so easy :
example 1 : 6x² - 2x = 2x(3x - 1)
example 2 : 21x - 7 = 7(3x - 1)
This polynominal becomes :
= 2x(3x - 1) + 7(3x - 1)
Fourth, fatorise a last time, it's evident : (3x - 1)
= (3x - 1)(2x + 7)
Therefore, you can write : 6x² + 19x - 7= (3x - 1)(2x + 7)
Set up possible factors of your a value, 6, and your c value, 7
factors of 6:
1, 6
2,3
Factors of -7:
1, -7
-1, 7
Play around with the correpsonding factors so that by multiplying them, they add up to the b value, 19
lets try 2, 3, and -1 7
so arrage them vertically to get:
3 -1
2 7
now cross multiply:
2*7 +(-1*2)=19. thus we know these are the factors of this polynomial,
therefore the answer is simply 6x^2 +19x -7 = (3x -1)(2x +7)
Hope this helps!
(7 + 2 x) (-1 + 3 x)
6x^2+19x-7 =
(2x + 7)(3x - 1)
You can't the only factors of -7 are 1 & and 7.
Also there is no common factor to pull out.
Hi,
Here is what I've learned in school:
Let's say you have the following equation: a*x^2+b*x+c=0 .
Calculate Delta = b^2 - 4*a*c
First solution: x1 = (-b + sqrt(Delta)) / (2*a)
Second solution: x2 = (-b - sqrt(Delta) / (2*a)
In your case: a=6, b=19, c=-7.
Delta = 529
x1 = 4/12 = 0.33
x2 = -42/12 = -3.5
The equation can be written as a*(x-x1)*(x-x2):
6*(x-4/12)*(x+42/12) = (3x-1)(2x+7)
6x^2 + 19x - 7
6x^2 + 21x - 2x - 7
(6x^2 + 21x) - (2x - 7)
3x(2x + 7) - 1(2x + 7)
(3x - 1)(2x + 7)