Solution
1. x =(12-√228)/2=6-√ 57 = -1.550
2. x =(12+√228)/2=6+√ 57 = 13.550
H3LL0 this is How to get this^ is down there i suggest you take some notesL0L
The first term is,
x2 its coefficient is 1 .
The middle term is, -12x its coefficient is -12 .
The last term, "the constant", is -21
Multiply the coefficient of the first term by the constant
1 • -21 = -21
Find two factors of -21 whose sum equals the coefficient of the middle term, which is -12 .
-21 + 1 = -20
-7 + 3 = -4
-3 + 7 = 4
-1 + 21 = 20
Find the Vertex of
y = x2-12x-21
For any parabola,
Ax2+Bx+C,the x
-coordinate of the vertex is given by -B/(2A) is 6.0000
Plugging into the parabola formula 6.0000.
for x
we can calculate the y -coordinate :
y = 1.0 * 6.00 * 6.00 - 12.0 * 6.00 - 21.0
or y = -57.000
Root plot for : y = x2-12x-21Axis of Symmetry (dashed) {x}={ 6.00}
Vertex at {x,y} = { 6.00,-57.00}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-1.55, 0.00}
Root 2 at
{x,y} = {13.55, 0.00}
read the squar
2.2 Solving
x2-12x-21 = 0 by Completing The Square .
Add 21 to both side of the equation : x2-12x = 21 Now the clever bit:
Take the coefficient of x
, which is 12 , divide by two, giving 6 , and finally square it giving 36
Adding
36 has completed the left hand side into a perfect square :
x2-12x+36 =
(x-6) • (x-6) =
(x-6)2
Since
x2-12x+36 = 57 and x2-12x+36 = (x-6)2
then, according to the law of transitivity,
(x-6)2 = 57
We'll refer to this Equation as
Eq. #2.2.1
The
Square Root Principle says that When two things are equal, their square roots are equal.
the square root of
(x-6)2 is (x-6)2/2 =
(x-6)1 =
x-6
Now, applying the Square Root Principle to Eq. #2.2.1 we get:
x-6 = √ 57
Add
6 to both sides to obtain: x = 6 + √ 57
Since a square root has two values, one positive and the other negative
x2 - 12x - 21 = 0 has two solutions:
x = 6 + √ 57 or
x = 6 - √ 57
using the Quadratic Formula
2.3 Solving x2-12x-21 = 0 by the Quadratic Formula .
According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
x = ————————
2A
In our case, A = 1
B = -12
C = -21
Accordingly, B2 - 4AC =
144 - (-84) =
228
Applying the quadratic formula :
12 ± √ 228
x = ——————
2
The prime factorization of 228 is
2•2•3•19
To be able to remove something from under the radical, there have to be
2 instances of it because we are taking a square
i.e. second root
√ 228 = √ 2•2•3•19 =
± 2 • √ 57 √ 57 , rounded to 4 decimal digits, is 7.5498
So now we are looking at:
x = ( 12 ± 2 • 7.550 ) / 2
Two real solutions:
x =(12+√228)/2=6+√ 57 = 13.550 or:
x =(12-√228)/2=6-√ 57 = -1.550
