Answer:
d = 2
Explanation:
You want the value of d that makes y=7 an extraneous solution to ...
√(4y -3) = d -y
Extraneous solution
A radical equation has an extraneous solution if it depends on the value of the radical being negative, rather than positive.
Here, that means ...
√(4·7 -3) = d -7
√25 +7 = d
If the value of the radical is -5, rather than +5, then we have ...
-5 +7 = d = 2
The value 2 for the constant d will make y=7 an extraneous solution.
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Additional comment
For d=2, the actual solution is y=1. For that value of d, y=7 is extraneous. The attachment shows how the extraneous solution depends on the negative branch of the square root curve.
If you solve this in the usual way, you will square both sides:
4y -3 = d² -2dy +y²
y² -(4+2d)y +3 +d² = 0
If y=7 is a solution to this, we must have ...
7² -(4+2d)·7 +3 +d² = 0
d² -14d +24 = 0
(d -2)(d -12) = 0
d = 2 or 12
For d = 12, y = 7 is the real solution, and y = 21 becomes an extraneous solution. The value d=2 makes y=7 an extraneous solution.
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