Respuesta :
Step-by-step explanation:
w² + 9w + 14 = 0
w² + 7w + 2w + 14 = 0
w(w + 7) + 2(w + 7) = 0
(w + 2) (w + 7) = 0
Thus, w = (-2) and (-7)
-2,-7
The common denominator of the two fractions is 4 Adding (-56/4)+(81/4) gives 25/4
So adding to both sides we finally get :
w2+9w+(81/4) = 25/4
Adding 81/4 has completed the left hand side into a perfect square :
w2+9w+(81/4) =
(w+(9/2)) • (w+(9/2)) =
(w+(9/2))2
Things which are equal to the same thing are also equal to one another. Since
w2+9w+(81/4) = 25/4 and
w2+9w+(81/4) = (w+(9/2))2
then, according to the law of transitivity,
(w+(9/2))2 = 25/4
We'll refer to this Equation as Eq. #3.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(w+(9/2))2 is
(w+(9/2))2/2 =
(w+(9/2))1 =
w+(9/2)
Now, applying the Square Root Principle to Eq. #3.2.1 we get:
w+(9/2) = √ 25/4
Subtract 9/2 from both sides to obtain:
w = -9/2 + √ 25/4
Since a square root has two values, one positive and the other negative
w2 + 9w + 14 = 0
has two solutions:
w = -9/2 + √ 25/4
or
w = -9/2 - √ 25/4
Note that √ 25/4 can be written as
√ 25 / √ 4 which is 5 / 2
Solve Quadratic Equation using the Quadratic Formula
3.3 Solving w2+9w+14 = 0 by the Quadratic Formula .
According to the Quadratic Formula, w , the solution for Aw2+Bw+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
w = ————————
2A
In our case, A = 1
B = 9
C = 14
Accordingly, B2 - 4AC =
81 - 56 =
25
Applying the quadratic formula :
-9 ± √ 25
w = —————
2
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:
w = ( -9 ± 5) / 2
Two real solutions:
w =(-9+√25)/2=(-9+5)/2= -2.000
or:
w =(-9-√25)/2=(-9-5)/2= -7.000
Two solutions were found :
w = -2
w = -7
