The solution to x^2-2x<1 is -sqrt(2)+1<x<sqrt(2)+1

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x^2-2x<1

Solution

x^{2} - 2 x < 1

- 2 + 1 < x < 2 + 1

Interval Notation

(- 2 + 1, 2 + 1)

Decimal Notation

- 0.41421 \dots < x < 2.41421 \dots

Solution steps

x^{2} - 2 x < 1

Rewrite in standard form

x^{2} - 2 x - 1 < 0

Complete the square

x^{2} - 2 x - 1 : (x - 1)^{2} - 2

(x - 1)^{2} - 2 < 0

Move

2

to the right side

(x - 1)^{2} < 2

For

u^{n} < a

, if

n

is even then

- 2 < x - 1 < 2

a < u < b

then

a < u and u < b

- 2 < x - 1 and x - 1 < 2

- 2 < x - 1 : x > - 2 + 1

x - 1 < 2 : x < 2 + 1

Combine the intervals

x > - 2 + 1 and x < 2 + 1

Merge Overlapping Intervals

- 2 + 1 < x < 2 + 1

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

(x + 2) \cdot x (x - 4) \leq 0

\frac{( x ^{2} - 3 x + 4 )}{( x + 5 )} < \frac{1}{3}

4 m^{2} + 4 m - 8 > 0

\frac{3}{( x ^{2} - x + 1 )} > 1