add 8y²+6y+3and -3y²+4y+9 and y²+8y
Answers
Answer:
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Step-by-step explanation:
Using a tried and true DS strategy, start with the easier statement, Statement #2.
Statement #2:
(y – 4)(y + 2) = 0
y = +4 or y = –2
Since there are two values of y, this statement, alone and by itself, is not sufficient.
Statement #1:
This is an equation with a radical. The radical is already isolated, so square both sides.
(3y – 1)2 = 8y2 – 4y + 9
9y2 – 6y + 1 = 8y2 – 4y + 9
y2 – 2y – 8 = 0
Lo and behold! We have arrived at the same equation we found in Statement #2, with solutions y = +4 or y = –2. The naïve conclusion would be—this statement says exactly the same thing as the other. That's incorrect, though, because we don't know whether both of these values are valid solutions, or whether one or more is an extraneous root. We need to test this in the original equation.
Test y = +4 on equation 3y-1=√(8y^2-4y+9)
LHS=RHS
Test y = –2 on equation 3y-1=√(8y^2-4y+9)
The LHS and RHS are not equal, so this does not check! This value, y = –2, is an extraneous root.
(NB: it's often the case that an extraneous root will make the two sides equal to values equal in absolute value and opposite in sign.)
Thus, the equation given in Statement #1 has only one solution, y = 4, so this equation provides a definitive answer to the prompt question. This statement, alone and by itself, is sufficient.
- B ± √ B2-4AC
y = ————————
2A
In our case, A = 8
B = -10
C = 3
Accordingly, B2 - 4AC =
100 - 96 =
4
Applying the quadratic formula :
10 ± √ 4
y = ————
16
Can √ 4 be simplified ?
Yes! The prime factorization of 4 is
2•2
√ 4 = √ 2•2 =
± 2 • √ 1 =
± 2
So now we are looking at:
y = ( 10 ± 2) / 16
Two real solutions:
y =(10+√4)/16=(5+)/8= 0.750
or:
y =(10-√4)/16=(5-)/8= 0.500