Solve root root 2 x^2-2x 1

Question

Simplify the expression

2 x - 2 x

Evaluate

(2 x)^{2} - 2 x \times 1

Evaluate the power

2 x - 2 x \times 1

Solution

2 x - 2 x

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Factor the expression

2 x \times (1 - 2 x)

Evaluate

(2 x)^{2} - 2 x \times 1

Simplify the root

More Steps

Evaluate

2 x

Use m n a = mn a to simplify the expression

2 \times 2 2 x

Multiply the numbers

4 2 x

(4 2 x)^{2} - 2 x \times 1

Multiply the terms

(4 2 x)^{2} - 2 x

Rewrite the expression

4 2 x \times 4 2 x - 4 2 x \times 4 8 x^{3}

Factor out 4 2 x from the expression

4 2 x \times (4 2 x - 4 8 x^{3})

Factor the expression

More Steps

Evaluate

4 2 x - 4 8 x^{3}

Rewrite the expression

4 2 x - 4 2 x \times 2 x

Factor out 4 2 x from the expression

4 2 x \times (1 - 2 x)

4 2 x \times 4 2 x \times (1 - 2 x)

Solution

2 x \times (1 - 2 x)

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Find the roots

x_{1} = 0, x_{2} = \frac{1}{2}

Alternative Form

x_{1} = 0, x_{2} = 0.5

Evaluate

(2 x)^{2} - 2 x \times 1

To find the roots of the expression,set the expression equal to 0

(2 x)^{2} - 2 x \times 1 = 0

Find the domain

More Steps

Evaluate

{2 x \geq 0 2 x \geq 0

Calculate

{x \geq 0 2 x \geq 0

Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x

{x \geq 0 x \in R

Find the intersection

x \geq 0

(2 x)^{2} - 2 x \times 1 = 0, x \geq 0

Calculate

(2 x)^{2} - 2 x \times 1 = 0

Simplify the root

More Steps

Evaluate

2 x

Use m n a = mn a to simplify the expression

2 \times 2 2 x

Multiply the numbers

4 2 x

(4 2 x)^{2} - 2 x \times 1 = 0

Multiply the terms

(4 2 x)^{2} - 2 x = 0

Rewrite the expression

2 x - 2 x = 0

Move the expression to the right-hand side and change its sign

2 x = 2 x

Evaluate

2 x = 2 x, 2 x \geq 0

Evaluate

2 x = 2 x, x \geq 0

Solve the equation for x

More Steps

Evaluate

2 x = 2 x

Raise both sides of the equation to the 2 -th power to eliminate the isolated 2 -th root

(2 x)^{2} = (2 x)^{2}

Evaluate the power

2 x = 4 x^{2}

Add or subtract both sides

2 x - 4 x^{2} = 0

Factor the expression

More Steps

Evaluate

2 x - 4 x^{2}

Rewrite the expression

2 x - 2 x \times 2 x

Factor out 2 x from the expression

2 x (1 - 2 x)

2 x (1 - 2 x) = 0

When the product of factors equals 0,at least one factor is 0

2 x = 0 1 - 2 x = 0

Solve the equation for x

x = 0 1 - 2 x = 0

Solve the equation for x

More Steps

Evaluate

1 - 2 x = 0

Move the constant to the right-hand side and change its sign

- 2 x = 0 - 1

Removing 0 doesn't change the value,so remove it from the expression

- 2 x = - 1

Change the signs on both sides of the equation

2 x = 1

Divide both sides

\frac{2 x}{2} = \frac{1}{2}

Divide the numbers

x = \frac{1}{2}

x = 0 x = \frac{1}{2}

x = 0 x = \frac{1}{2}, x \geq 0

Find the intersection of the solution and the defined range

x = 0 x = \frac{1}{2}

Solution

x_{1} = 0, x_{2} = \frac{1}{2}

Alternative Form

x_{1} = 0, x_{2} = 0.5

Show Solution