Solve sqrt(x - 1) sqrt(3x - 3) = sqrt(x ^ 2 - 1)

Question

Solve the equation

x_{1} = 1, x_{2} = 2

Evaluate

x - 1 \times 3 x - 3 = x^{2} - 1

Find the domain

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Evaluate

⎩ ⎨ ⎧ x - 1 \geq 0 3 x - 3 \geq 0 x^{2} - 1 \geq 0

Calculate

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Evaluate

x - 1 \geq 0

Move the constant to the right side

x \geq 0 + 1

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

x \geq 1

⎩ ⎨ ⎧ x \geq 1 3 x - 3 \geq 0 x^{2} - 1 \geq 0

Calculate

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Evaluate

3 x - 3 \geq 0

Move the constant to the right side

3 x \geq 0 + 3

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

3 x \geq 3

Divide both sides

\frac{3 x}{3} \geq \frac{3}{3}

Divide the numbers

x \geq \frac{3}{3}

Divide the numbers

x \geq 1

⎩ ⎨ ⎧ x \geq 1 x \geq 1 x^{2} - 1 \geq 0

Calculate

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Evaluate

x^{2} - 1 \geq 0

Move the constant to the right side

x^{2} \geq 1

Take the 2 -th root on both sides of the inequality

x^{2} \geq 1

Calculate

∣ x ∣ \geq 1

Separate the inequality into 2 possible cases

x \geq 1 x \leq - 1

Find the union

x \in (- \infty, - 1] \cup [1, + \infty)

⎩ ⎨ ⎧ x \geq 1 x \geq 1 x \in (- \infty, - 1] \cup [1, + \infty)

Simplify

{x \geq 1 x \in (- \infty, - 1] \cup [1, + \infty)

Find the intersection

x \geq 1

x - 1 \times 3 x - 3 = x^{2} - 1, x \geq 1

Multiply the terms

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Evaluate

x - 1 \times 3 x - 3

The product of roots with the same index is equal to the root of the product

(x - 1) (3 x - 3)

Calculate the product

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Evaluate

(x - 1) (3 x - 3)

Apply the distributive property

x \times 3 x - x \times 3 - 3 x - (- 3)

Multiply the terms

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

Use the commutative property to reorder the terms

3 x^{2} - 3 x - 3 x - (- 3)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

3 x^{2} - 3 x - 3 x + 3

Subtract the terms

3 x^{2} - 6 x + 3

3 x^{2} - 6 x + 3

Factor the expression

3 (x - 1)^{2}

The root of a product is equal to the product of the roots of each factor

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

Reduce the index of the radical and exponent with 2

3 \times (x - 1)

Apply the distributive property

3 \times x - 3 \times 1

Any expression multiplied by 1 remains the same

3 \times x - 3

3 \times x - 3 = x^{2} - 1

Swap the sides

x^{2} - 1 = 3 \times x - 3

Evaluate

x^{2} - 1 = 3 \times x - 3, 3 \times x - 3 \geq 0

Evaluate

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Evaluate

3 \times x - 3 \geq 0

Move the constant to the right side

3 \times x \geq 0 + 3

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

3 \times x \geq 3

Divide both sides

\frac{3 \times x}{3} \geq \frac{3}{3}

Divide the numbers

x \geq \frac{3}{3}

Divide the numbers

x \geq 1

x^{2} - 1 = 3 \times x - 3, x \geq 1

Solve the equation for x

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Evaluate

x^{2} - 1 = 3 \times x - 3

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

(x^{2} - 1)^{2} = (3 \times x - 3)^{2}

Evaluate the power

x^{2} - 1 = 3 x^{2} - 6 x + 3

Move the expression to the left side

x^{2} - 1 - (3 x^{2} - 6 x + 3) = 0

Subtract the terms

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Evaluate

x^{2} - 1 - (3 x^{2} - 6 x + 3)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

x^{2} - 1 - 3 x^{2} + 6 x - 3

Subtract the terms

- 2 x^{2} - 1 + 6 x - 3

Subtract the numbers

- 2 x^{2} - 4 + 6 x

- 2 x^{2} - 4 + 6 x = 0

Factor the expression

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Evaluate

- 2 x^{2} - 4 + 6 x

Evaluate

- 2 x^{2} + 6 x - 4

Rewrite the expression

- 2 x^{2} + 2 \times 3 x - 2 \times 2

Factor out - 2 from the expression

- 2 (x^{2} - 3 x + 2)

Factor the expression

- 2 (x - 2) (x - 1)

- 2 (x - 2) (x - 1) = 0

Divide the terms

(x - 2) (x - 1) = 0

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

x - 2 = 0 x - 1 = 0

Solve the equation for x

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Evaluate

x - 2 = 0

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

x = 0 + 2

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

x = 2

x = 2 x - 1 = 0

Solve the equation for x

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Evaluate

x - 1 = 0

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

x = 0 + 1

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

x = 1

x = 2 x = 1

x = 2 x = 1, x \geq 1

Find the intersection of the solution and the defined range

x = 2 x = 1

Check the solution

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Check the solution

2 - 1 \times 3 \times 2 - 3 = 2^{2} - 1

Simplify

3 = 3

Evaluate

true

x = 2 x = 1

Check the solution

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Check the solution

1 - 1 \times 3 \times 1 - 3 = 1^{2} - 1

Simplify

0 = 0

Evaluate

true

x = 2 x = 1

Solution

x_{1} = 1, x_{2} = 2

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