Question

Which equation has the solutions x = 1 plus-or-minus StartRoot 5

Answer 1

For this case we have the following solutions:

$x_ {1} = 1 + \sqrt {5}\\x_ {2} = 1- \sqrt {5}$

Then, the factorized equation is of the form:

$(x- (1+ \sqrt {5})) (x- (1- \sqrt {5})) = 0\\(x-a) (x-b) = 0$

We apply distributive property:

$x ^ 2-bx-ax + ab = 0\\x ^ 2- (1- \sqrt {5}) x- (1+ \sqrt {5}) x + (1+ \sqrt {5}) (1- \sqrt {5}) = 0\\x ^ 2-x + \sqrt {5} x-x- \sqrt {5} x + (1 ^ 2- \sqrt {5} + \sqrt {5} - (\sqrt {5}) ^ 2) = 0\\x ^ 2-2x + (1-5) = 0\\x ^ 2-2x-4 = 0$

ANswer:

The equation is $x ^ 2-2x-4$