Answer:
x =(6-√108)/2=3-3√ 3 = -2.196
x =(6+√108)/2=3+3√ 3 = 8.196
Step-by-step explanation:
Step 1 :
Equation at the end of step 1 :
2 • (x - 3)2 - 54 = 0
Step 2 :
2.1 Evaluate : (x-3)2 = x2-6x+9
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
2x2 - 12x - 36 = 2 • (x2 - 6x - 18)
Adding 9 has completed the left hand side into a perfect square :
x2-6x+9 =
(x-3) • (x-3) =
(x-3)2 (x-3)1 =
x-3
Now, applying the Square Root Principle to Eq. #4.3.1 we get:
x-3 = √ 27
Add 3 to both sides to obtain:
x = 3 + √ 27
Since a square root has two values, one positive and the other negative
x2 - 6x - 18 = 0
has two solutions:
x = 3 + √ 27
or
x = 3 - √ 27
Solve Quadratic Equation using the Quadratic Formula
4.4 Solving x2-6x-18 = 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 = -6
C = -18
Accordingly, B2 - 4AC =
36 - (-72) =
108
Applying the quadratic formula :
6 ± √ 108
x = —————
2
Can √ 108 be simplified ?
Yes! The prime factorization of 108 is
2•2•3•3•3
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).
√ 108 = √ 2•2•3•3•3 =2•3•√ 3 =
± 6 • √ 3
√ 3 , rounded to 4 decimal digits, is 1.7321
So now we are looking at:
x = ( 6 ± 6 • 1.732 ) / 2
Two real solutions:
x =(6+√108)/2=3+3√ 3 = 8.196
or:
x =(6-√108)/2=3-3√ 3 = -2.196
