Solve by completing the square method:5y2-2y-2=0
Answers
Answer:
5y2-2y-2=0
Two solutions were found :
y =(2-√44)/10=(1-√ 11 )/5= -0.463
y =(2+√44)/10=(1+√ 11 )/5= 0.863
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(5y2 - 2y) - 2 = 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 5y2-2y-2
The first term is, 5y2 its coefficient is 5 .
The middle term is, -2y its coefficient is -2 .
The last term, "the constant", is -2
Step-1 : Multiply the coefficient of the first term by the constant 5 • -2 = -10
Step-2 : Find two factors of -10 whose sum equals the coefficient of the middle term, which is -2 .
-10 + 1 = -9
-5 + 2 = -3
-2 + 5 = 3
-1 + 10 = 9
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 2 :
5y2 - 2y - 2 = 0
Step 3 :
Parabola, Finding the Vertex :
3.1 Find the Vertex of t = 5y2-2y-2
Step-by-step explanation:
Since a square root has two values, one positive and the other negative
y2 - (2/5)y - (2/5) = 0
has two solutions:
y = 1/5 + √ 11/25
or
y = 1/5 - √ 11/25
Note that √ 11/25 can be written as
√ 11 / √ 25 which is √ 11 / 5
Solve Quadratic Equation using the Quadratic Formula
3.3 Solving 5y2-2y-2 = 0 by the Quadratic Formula .
According to the Quadratic Formula, y , the solution for Ay2+By+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
y = ————————
2A
In our case, A = 5
B = -2
C = -2
Accordingly, B2 - 4AC =
4 - (-40) =
44
Applying the quadratic formula :
2 ± √ 44
y = —————
10
Can √ 44 be simplified ?
Yes! The prime factorization of 44 is
2•2•11
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).
√ 44 = √ 2•2•11 =
± 2 • √ 11
√ 11 , rounded to 4 decimal digits, is 3.3166
So now we are looking at:
y = ( 2 ± 2 • 3.317 ) / 10
Two real solutions:
y =(2+√44)/10=(1+√ 11 )/5= 0.863
or:
y =(2-√44)/10=(1-√ 11 )/5= -0.463
Two solutions were found :
y =(2-√44)/10=(1-√ 11 )/5= -0.463
y =(2+√44)/10=(1+√ 11 )/5= 0.863
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