Answer: 7
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Explanation:
Let,
[tex]A = \sqrt{14y^2 - 20y + 48}\\\\B = \sqrt{14y^2 - 20y - 15}\\\\[/tex]
The first equation
[tex]\sqrt{14y^2 - 20y + 48} + \sqrt{14y^2 - 20y - 15} = 9[/tex]
is in the form of
[tex]A+B = 9[/tex]
The goal is to find the value of
[tex]\sqrt{14y^2 - 20y + 48} - \sqrt{14y^2 - 20y -15}[/tex]
which is of the form [tex]A - B[/tex]
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Use the difference of squares rule to say the following:
[tex](A+B)(A-B) = A^2 - B^2\\\\(A+B)(A-B) = \left(\sqrt{14y^2 - 20y + 48}\right)^2 -\left(\sqrt{14y^2 - 20y - 15 }\right)^2\\\\(A+B)(A-B) = (14y^2 - 20y + 48) -(14y^2 - 20y - 15 )\\\\(A+B)(A-B) = 14y^2 - 20y + 48 -14y^2 + 20y + 15 \\\\(A+B)(A-B) = (14y^2 -14y^2) +(- 20y + 20y) + (48+15) \\\\(A+B)(A-B) = 63 \\\\[/tex]
All of the y^2 and y terms cancel, leaving nothing but a single number.
The last thing to do is replace the A+B with 9, since A+B = 9 was mentioned earlier, and isolate A-B
Divide both sides by 9 to isolate A-B
[tex](A+B)(A-B) = 63 \\\\9(A-B) = 63 \\\\A-B = 63/9 \\\\A-B = 7 \\\\\sqrt{14y^2 - 20y + 48} - \sqrt{14y^2 - 20y - 15} = 7 \\\\[/tex]
Therefore, 7 is the final answer.
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An alternative is to solve the first equation for y.
It's messy but doable. I'll skip steps, but you should get y = -4/7 and y = 2 as the two solutions.
If you plugged y = -4/7 into the second expression, then you should end up with 7. The same goes for plugging in y = 2. Only pick one of those values.
