If x + 2y = 9 and xy = 7, the value of x2 + 4y2 is
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Answers
Xy= 7
x+2y= 9
Therefore,
Y= 2 and x=3 as
x+2y=9
3+ (2×2)=9
therefore,
X2 + 4y2 =
6 + 16= 22
Answer= 22
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x^2 + 4y^2 = 17
xy = 2
Solve for x in the second equation:
x = 2/y
Now substitute for x in the second equation:
x^2 + 4y^2 = 17
x = 2/y
(2/y)(2/y) + 4y^2 = 17
4/y^2 + 4y^2 = 17
Multiply all terms by y^2 to eliminate the denominator:
4 + 4y^4 = 17y^2
Subtract 17y^2 from both sides:
4y^4 - 17y^2 + 4 = 0
Factor:
(4y^2–1)(y^2-4) = 0
There are four possible solutions to this equation, and the way to solve them is to first solve each set of parentheses by setting it equal to zero. We do this because when one set of parentheses is multiplied by the other set the product will be zero and thus this equation will be a true statement.
We’ll begin with the binomial on the left:
4y^2–1 = 0
Add 1 to both sides:
4y^2 = 1
Divide by 4:
y^2 = 1/4
The square root of 1/4 is +/- 1/2, so
y = +/- 1/2
Now for the other set of parentheses:
y^2 - 4 = 0
Add 4 to both sides:
y^2 = 4
the square root of 4 is +/-2, so
y = +/-2
Thus, we have four solutions for y:
y = 1/2
y = -1/2
y = 2
y = -2
and we need to test for each value of y in the original equations:
x^2 + 4y^2 = 17
xy = 2
And remember that:
x = 2/y
which makes this:
x^2 + 4y^2 = 17
into this:
4/y^2 + 4y^2 = 17
We’ll start with y = 1/2 first:
4/y^2 + 4y^2 = 17
y = 1/2
4/(1/4) + 4/4 = 17
Dividing by 1/4 is the same as multiplying by 4:
4 * 4 + 4/4 = 17
16 + 1 = 17
17 = 17
So this solution works! Next solution:
y = -1/2
4/-(1/2)^2 + 4*-(1/2)^2 = 17
4/1/4 + 4*1/4 = 17
4*4 + 4/4 = 17
16 + 1 = 17
17 = 17
Okay! Two solutions down, two to go:
y = 2
4/y^2 + 4y^2 = 17
y = 2
4/4 + 4*4 = 17
1 + 16 = 17
17 = 17
One more value!
y = -2
4/(-2)^2 + 4*(-2)^2 = 17
4/4 + 16 = 17
1 + 16 = 17
17 = 17
Last one!
y = 2
4/2^2 + 4*2^2 = 17
4/4 + 4*4 = 17
1 + 16 = 17
17 = 17
So all four values we have for y are solutions to the first two equations.
Now, for the final equation:
x + 2y
x = 2/y
Again, y has four solutions, so we can come up with four different values for this expression.
x = 2/y
y = 1/2
2/(1/2) + 2*1/2
2*2 + 2/2
4+1
5
y = (-1/2)
2/(-1/2) + 2*-1/2
2*-2 - 2/2
-4 - 1
-5
y = 2
2/2 + 2*2
1 + 4
5
So there are two different values for the expression x + 2y: x + 2y = 5 OR x + 2y = -5
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