+109 Find the greatest value of "a" such that \(\frac{7\sqrt{\left(2a\right)^2+\left(1\right)^2}-4a^2-1}{\sqrt{1+4a^2}+3}=2\)
Solve for a:
(-1 - 4 a^2 + 7 sqrt(4 a^2 + 1))/(sqrt(4 a^2 + 1) + 3) = 2
Multiply both sides by sqrt(4 a^2 + 1) + 3:
-1 - 4 a^2 + 7 sqrt(4 a^2 + 1) = 2 (sqrt(4 a^2 + 1) + 3)
Add 4 a^2 + 1 to both sides:
7 sqrt(4 a^2 + 1) = 1 + 4 a^2 + 2 (sqrt(4 a^2 + 1) + 3)
Subtract 1 + 4 a^2 + 2 (sqrt(4 a^2 + 1) + 3) from both sides:
-1 - 4 a^2 + 7 sqrt(4 a^2 + 1) - 2 (sqrt(4 a^2 + 1) + 3) = 0
-1 - 4 a^2 + 7 sqrt(4 a^2 + 1) - 2 (sqrt(4 a^2 + 1) + 3) = -7 - 4 a^2 + 5 sqrt(4 a^2 + 1):
-7 - 4 a^2 + 5 sqrt(4 a^2 + 1) = 0
Simplify and substitute x = sqrt(4 a^2 + 1).
-7 - 4 a^2 + 5 sqrt(4 a^2 + 1) = -6 + 5 sqrt(4 a^2 + 1) - (sqrt(4 a^2 + 1))^2
= -x^2 + 5 x - 6:
-x^2 + 5 x - 6 = 0
The left hand side factors into a product with three terms:
-(x - 3) (x - 2) = 0
Multiply both sides by -1:
(x - 3) (x - 2) = 0
Split into two equations:
x - 3 = 0 or x - 2 = 0
Add 3 to both sides:
x = 3 or x - 2 = 0
Substitute back for x = sqrt(4 a^2 + 1):
sqrt(4 a^2 + 1) = 3 or x - 2 = 0
Raise both sides to the power of two:
4 a^2 + 1 = 9 or x - 2 = 0
Subtract 1 from both sides:
4 a^2 = 8 or x - 2 = 0
Divide both sides by 4:
a^2 = 2 or x - 2 = 0
Take the square root of both sides:
a = sqrt(2) or a = -sqrt(2) or x - 2 = 0
Add 2 to both sides:
a = sqrt(2) or a = -sqrt(2) or x = 2
Substitute back for x = sqrt(4 a^2 + 1):
a = sqrt(2) or a = -sqrt(2) or sqrt(4 a^2 + 1) = 2
Raise both sides to the power of two:
a = sqrt(2) or a = -sqrt(2) or 4 a^2 + 1 = 4
Subtract 1 from both sides:
a = sqrt(2) or a = -sqrt(2) or 4 a^2 = 3
Divide both sides by 4:
a = sqrt(2) or a = -sqrt(2) or a^2 = 3/4
Take the square root of both sides:
a = sqrt(2) or a = -sqrt(2) or a = sqrt(3)/2 or a = -sqrt(3)/2
The greatest value of "a" =Sqrt(2)
