Find the exact answer 4x x2 12Solution4 x x2 12 22 x x

Find the exact answer 4^x - x^2 = 1/2

Solution

4 ^{x - x^2} = 1/2

==> (2²) ^{x - x^2} = 2^-1           since 1/aⁿ = a^-n

==> 2^{2(x - x^2)} = 2^-1              since (a^m)ⁿ = a^mn

==> 2^{2x - 2x^2} = 2^-1

bases are equal ==> their powers can be equated

==> 2x - 2x² = -1

==> 2x² - 2x - 1 = 0

==> 2(x² - x) - 1 = 0

==> 2(x² - 2(x)(1/2) ) - 1 =0

==> 2(x² - 2(x)(1/2) + (1/2)² - (1/2)²) - 1 = 0

==> 2(x - 1/2)² - 1/2 - 1 = 0               since (a² - 2ab + b²) = (a - b)²

==> 2(x - 1/2)² = 3/2

==> (x - 1/2)² = 3/4

==> x - 1/2 = 3 /2 , - 3 /2

==> x = 1/2 + 3 /2 , 1/2 - 3 /2

==> x = (1 + 3)/2 , (1 - 3)/2

Hence solutions to given equality are x = (1 + 3)/2 , (1 - 3)/2