2x2-7x+3 = 0
Divide both sides of the equation by 2 to have 1 as the coefficient of the first term :
x2-(7/2)x+(3/2) = 0
Subtract 3/2 from both side of the equation :
x2-(7/2)x = -3/2
Now the clever bit: Take the coefficient of x , which is 7/2 , divide by two, giving 7/4 , and finally square it giving 49/16
Add 49/16 to both sides of the equation :
On the right hand side we have :
-3/2 + 49/16 The common denominator of the two fractions is 16 Adding (-24/16)+(49/16) gives 25/16
So adding to both sides we finally get :
x2-(7/2)x+(49/16) = 25/16
Adding 49/16 has completed the left hand side into a perfect square :
x2-(7/2)x+(49/16) =
(x-(7/4)) • (x-(7/4)) =
(x-(7/4))2
Things which are equal to the same thing are also equal to one another. Since
x2-(7/2)x+(49/16) = 25/16 and
x2-(7/2)x+(49/16) = (x-(7/4))2
then, according to the law of transitivity,
(x-(7/4))2 = 25/16
We'll refer to this Equation as Eq. #6.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-(7/4))2 is
(x-(7/4))2/2 =
(x-(7/4))1 =
x-(7/4)
Now, applying the Square Root Principle to Eq. #6.2.1 we get:
x-(7/4) = √ 25/16
Add 7/4 to both sides to obtain:
x = 7/4 + √ 25/16
Since a square root has two values, one positive and the other negative
x2 - (7/2)x + (3/2) = 0
has two solutions:
x = 7/4 + √ 25/16
or
x = 7/4 - √ 25/16
Note that √ 25/16 can be written as
√ 25 / √ 16 which is 5 / 4
Solve Quadratic Equation using the Quadratic Formula
6.3 Solving 2x2-7x+3 = 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 = 2
B = -7
C = 3
Accordingly, B2 - 4AC =
49 - 24 =
25
Applying the quadratic formula :
7 ± √ 25
x = —————
4
Can √ 25 be simplified ?
Yes! The prime factorization of 25 is
5•5
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).
√ 25 = √ 5•5 =
± 5 • √ 1 =
± 5
So now we are looking at:
x = ( 7 ± 5) / 4
Two real solutions:
x =(7+√25)/4=(7+5)/4= 3.000
or:
x =(7-√25)/4=(7-5)/4= 0.500
Two solutions were found :
x = 1/2 = 0.500
x = 3
