2×2-2x-1=0

Use the quadratic formula to find the solutions.

-b±b2-4(ac)2a

Substitute the values a=2, b=-2, and c=-1 into the quadratic formula and solve for x.

2±(-2)2-4⋅(2⋅-1)2⋅2

Simplify the numerator.

Raise -2 to the power of 2.

x=2±4-4⋅(2⋅-1)2⋅2

Multiply 2 by -1.

x=2±4-4⋅-22⋅2

Multiply -4 by -2.

x=2±4+82⋅2

Add 4 and 8.

x=2±122⋅2

Rewrite 12 as 22⋅3.

Factor 4 out of 12.

x=2±4(3)2⋅2

Rewrite 4 as 22.

x=2±22⋅32⋅2

x=2±22⋅32⋅2

Pull terms out from under the radical.

x=2±232⋅2

x=2±232⋅2

Multiply 2 by 2.

x=2±234

Simplify 2±234.

x=1±32

x=1±32

Simplify the numerator.

Raise -2 to the power of 2.

x=2±4-4⋅(2⋅-1)2⋅2

Multiply 2 by -1.

x=2±4-4⋅-22⋅2

Multiply -4 by -2.

x=2±4+82⋅2

Add 4 and 8.

x=2±122⋅2

Rewrite 12 as 22⋅3.

Factor 4 out of 12.

x=2±4(3)2⋅2

Rewrite 4 as 22.

x=2±22⋅32⋅2

x=2±22⋅32⋅2

Pull terms out from under the radical.

x=2±232⋅2

x=2±232⋅2

Multiply 2 by 2.

x=2±234

Simplify 2±234.

x=1±32

Change the ± to +.

x=1+32

x=1+32

Simplify the numerator.

Raise -2 to the power of 2.

x=2±4-4⋅(2⋅-1)2⋅2

Multiply 2 by -1.

x=2±4-4⋅-22⋅2

Multiply -4 by -2.

x=2±4+82⋅2

Add 4 and 8.

x=2±122⋅2

Rewrite 12 as 22⋅3.

Factor 4 out of 12.

x=2±4(3)2⋅2

Rewrite 4 as 22.

x=2±22⋅32⋅2

x=2±22⋅32⋅2

Pull terms out from under the radical.

x=2±232⋅2

x=2±232⋅2

Multiply 2 by 2.

x=2±234

Simplify 2±234.

x=1±32

Change the ± to -.

x=1-32

x=1-32

The final answer is the combination of both solutions.

x=1+32,1-32

The result can be shown in multiple forms.

Exact Form:

x=1+32,1-32

Decimal Form:

x=1.36602540…,-0.36602540…

Solve Using the Square Root Property 2x^2-2x-1=0