0.3
Step-by-step explanation: Solving 3x2-25x+8 = 0 by Completing The Square .
Divide both sides of the equation by 3 to have 1 as the coefficient of the first term :
x2-(25/3)x+(8/3) = 0
Subtract 8/3 from both side of the equation :
x2-(25/3)x = -8/3
Now the clever bit: Take the coefficient of x , which is 25/3 , divide by two, giving 25/6 , and finally square it giving 625/36
Add 625/36 to both sides of the equation :
On the right hand side we have :
-8/3 + 625/36 The common denominator of the two fractions is 36 Adding (-96/36)+(625/36) gives 529/36
So adding to both sides we finally get :
x2-(25/3)x+(625/36) = 529/36
Adding 625/36 has completed the left hand side into a perfect square :
x2-(25/3)x+(625/36) =
(x-(25/6)) • (x-(25/6)) =
(x-(25/6))2
Things which are equal to the same thing are also equal to one another. Since
x2-(25/3)x+(625/36) = 529/36 and
x2-(25/3)x+(625/36) = (x-(25/6))2
then, according to the law of transitivity,
(x-(25/6))2 = 529/36
We'll refer to this Equation as Eq. #4.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-(25/6))2 is
(x-(25/6))2/2 =
(x-(25/6))1 =
x-(25/6)
Now, applying the Square Root Principle to Eq. #4.2.1 we get:
x-(25/6) = √ 529/36
Add 25/6 to both sides to obtain:
x = 25/6 + √ 529/36
Since a square root has two values, one positive and the other negative
x2 - (25/3)x + (8/3) = 0
has two solutions:
x = 25/6 + √ 529/36
or
x = 25/6 - √ 529/36
Note that √ 529/36 can be written as √ 529 / √ 36 which is 23 / 6
Solving 3x2-25x+8 = 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 = 3
B = -25
C = 8
Accordingly, B2 - 4AC =
625 - 96 =
529
Applying the quadratic formula :
25 ± √ 529
x = ——————
6
Can √ 529 be simplified ?
Yes! The prime factorization of 529 is
23•23
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).
√ 529 = √ 23•23 =
± 23 • √ 1 =
± 23
So now we are looking at:
x = ( 25 ± 23) / 6
Two real solutions:
x =(25+√529)/6=(25+23)/6= 8.000
or:
x =(25-√529)/6=(25-23)/6= 0.333
