Algebra Examples

$x = 2 \sqrt{x - 1}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 \sqrt{x - 1} = x$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{x - 1})}^{2} = x^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 1}$ as ${(x - 1)}^{\frac{1}{2}}$ .

${(2 {(x - 1)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(2 {(x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $2 {(x - 1)}^{\frac{1}{2}}$ .

$2^{2} {({(x - 1)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.2.1.2

Raise $2$ to the power of $2$ .

$4 {({(x - 1)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.2.1.3

Multiply the exponents in ${({(x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 {(x - 1)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$4 {(x - 1)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$4 {(x - 1)}^{1} = x^{2}$

Step 3.2.1.4

Simplify.

$4 (x - 1) = x^{2}$

Step 3.2.1.5

Apply the distributive property.

$4 x + 4 \cdot - 1 = x^{2}$

Step 3.2.1.6

Multiply $4$ by $- 1$ .

$4 x - 4 = x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $x^{2}$ from both sides of the equation.

$4 x - 4 - x^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor $- 1$ out of $4 x - 4 - x^{2}$ .

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $- 4$ .

$4 x - x^{2} - 4 = 0$

Step 4.2.1.1.2

Reorder $4 x$ and $- x^{2}$ .

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

Step 4.2.1.2

Factor $- 1$ out of $- x^{2}$ .

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

Step 4.2.1.3

Factor $- 1$ out of $4 x$ .

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

Step 4.2.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 4.2.1.5

Factor $- 1$ out of $- (x^{2}) - (- 4 x)$ .

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

Step 4.2.1.6

Factor $- 1$ out of $- (x^{2} - 4 x) - 1 (4)$ .

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

Step 4.2.2

Factor using the perfect square rule.

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Step 4.2.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 4.2.2.3

Rewrite the polynomial.

$- (x^{2} - 2 \cdot x \cdot 2 + 2^{2}) = 0$

Step 4.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 4.3

Divide each term in $- {(x - 2)}^{2} = 0$ by $- 1$ and simplify.

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Step 4.3.1

Divide each term in $- {(x - 2)}^{2} = 0$ by $- 1$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

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

Step 4.3.2.2

Divide ${(x - 2)}^{2}$ by $1$ .

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

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $0$ by $- 1$ .

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

Step 4.4

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.5

Add $2$ to both sides of the equation.

$x = 2$