Question
Download Solution PDFIf x3 = 184 + y3 and x = 4 + y, then the value of (x + y) is (given that x > 0 and y > 0)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
x3 = 184 + y3
x = 4 + y
x > 0 and y > 0
Calculation:
Substitute x = 4 + y into x3 = 184 + y3
(4 + y)3 = 184 + y3
Expand (4 + y)3 :
(4 + y)3 = 43 + 3 × 42 × y + 3 × 4 × y2 + y3
64 + 48y + 12y2 + y3 = 184 + y3
Subtract y3 from both sides:
64 + 48y + 12y2 = 184
Rearrange the equation:
12y2 + 48y + 64 = 184
Subtract 184 from both sides:
12y2 + 48y + 64 - 184 = 0
12y2 + 48y - 120 = 0
Divide by 12:
y2 + 4y - 10 = 0
Solve the quadratic equation y2 + 4y - 10 = 0 using the quadratic formula:
\(y = \frac{-b \pm √{b^2 - 4ac}}{2a}\)
where a = 1 , b = 4 , and c = -10
\(y = \frac{-4 \pm √{4^2 - 4 × 1 × (-10)}}{2 × 1}\)
\(y = \frac{-4 \pm √{16 + 40}}{2}\)
\(y = \frac{-4 \pm √{56}}{2}\)
\(y = \frac{-4 \pm 2√{14}}{2}\)
\(y = -2 \pm √{14}\)
Since y > 0 , we take the positive root:
y = -2 + √14
Now, find x :
x = 4 + y
x = 4 + (-2 + √14)
x = 2 + √14
Finally, find x + y :
x + y = (2 + √14) + (-2 + √14)
x + y = 2√14
The value of x + y is 2√14.
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