Can some one please check my solution to dot product Q plzzzzzzzzzzzzzzzz

Given that a=(λi - j) and b=(4i - 3j) and the angle between a and b is 45°,find the value of λ

MY SOLUTION
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a.b=(λi - j).(4i - 3j)
----=4λ + 3
|a|=√(λ² + (-1)² ) and |b|=√(4² + 3²) and cos45°= 1/√2
======================================…
now
a.b = |a| |b| cosα
==> 4λ + 3 =√(λ² + (-1)²) * √(4² + 3²) * 1/√2
-------4λ + 3 =5 * √(λ² + 1) * 1/√2
--√2(4λ + 3)=5√(λ² + 1)
squaring both sides gives
2( 16λ² + 24λ + 9) = 25(λ² + 1)
and so
7λ² +48λ - 7= 0
======================================…
the answer to this question is (1/7) which i don't get from solving the quadratic also a quadratic gives two values where this a 1 which is another fact that i did crap

can someone please help me if i made an error it would be very very appreciated
thank you :)

Here,
a=(λi - j) and b=(4i - 3j), SO,
a.b = |a||b|cos(45)
(λi - j).(4i - 3j) = sqrt[λ^2+1].sqrt[4^2+3^2]cos(45)
4λ+3 = sqrt(25λ^2+25)/sqrt(2),

squaring both-sides,
16λ^2 +24λ +9 = (25λ^2 + 25)/2
7λ^2 +48λ -7 = 0,

(λ+7)(7λ-1) = 0
SO,
λ+7=0, ==> λ=-7 .............. neglected,
OR,
7λ -1 = 0, ==> 7λ = 1,
Hence,
λ = 1/7 >==========================< ANSWER