Answer:
1.) x = -4 OR
2.) x = -1/2 = -0.500
Explanation:
4.2 Solving 2x2+9x+4 = 0 by Completing The Square .
Divide both sides of the equation by 2 to have 1 as the coefficient of the first term :
x2+(9/2)x+2 = 0
Subtract 2 from both side of the equation :
x2+(9/2)x = -2
Now the clever bit: Take the coefficient of x , which is 9/2 , divide by two, giving 9/4 , and finally square it giving 81/16
Add 81/16 to both sides of the equation :
On the right hand side we have :
-2 + 81/16 or, (-2/1)+(81/16)
The common denominator of the two fractions is 16 Adding (-32/16)+(81/16) gives 49/16
So adding to both sides we finally get :
x2+(9/2)x+(81/16) = 49/16
Adding 81/16 has completed the left hand side into a perfect square :
x2+(9/2)x+(81/16) =
(x+(9/4)) • (x+(9/4)) =
(x+(9/4))2
Things which are equal to the same thing are also equal to one another. Since
x2+(9/2)x+(81/16) = 49/16 and
x2+(9/2)x+(81/16) = (x+(9/4))2
then, according to the law of transitivity,
(x+(9/4))2 = 49/16
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+(9/4))2 is
(x+(9/4))2/2 =
(x+(9/4))1 =
x+(9/4)
Now, applying the Square Root Principle to Eq. #4.2.1 we get:
x+(9/4) = √ 49/16
Subtract 9/4 from both sides to obtain:
x = -9/4 + √ 49/16
Since a square root has two values, one positive and the other negative
x2 + (9/2)x + 2 = 0
has two solutions:
x = -9/4 + √ 49/16
or
x = -9/4 - √ 49/16
Note that √ 49/16 can be written as
√ 49 / √ 16 which is 7 / 4
Solve Quadratic Equation using the Quadratic Formula
4.3 Solving 2x2+9x+4 = 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 = 9
C = 4
Accordingly, B2 - 4AC =
81 - 32 =
49
Applying the quadratic formula :
-9 ± √ 49
x = —————
4
Can √ 49 be simplified ?
Yes! The prime factorization of 49 is
7•7
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).
√ 49 = √ 7•7 =
± 7 • √ 1 =
± 7
So now we are looking at:
x = ( -9 ± 7) / 4
Two real solutions:
x =(-9+√49)/4=(-9+7)/4= -0.500
or:
x =(-9-√49)/4=(-9-7)/4= -4.000
Two solutions were found :
x = -4
x = -1/2 = -0.500
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