Algebra Examples

$\sqrt{4 x - 4} = x - 1$

Step 1

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

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

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x - 4}$ as ${(4 x - 4)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Multiply the exponents in ${({(4 x - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$4 x - 4 = (x - 1) (x - 1)$

Step 2.3.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$4 x - 4 = x (x - 1) - 1 (x - 1)$

Step 2.3.1.2.2

Apply the distributive property.

$4 x - 4 = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Step 2.3.1.2.3

Apply the distributive property.

$4 x - 4 = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $x$ by $x$ .

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

Step 2.3.1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 2.3.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.3.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.3.1.3.2

Subtract $x$ from $- x$ .

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

Step 3

Solve for $x$ .

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Step 3.1

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

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

Step 3.2

Move all terms containing $x$ to the left side of the equation.

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Step 3.2.1

Subtract $4 x$ from both sides of the equation.

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

Step 3.2.2

Subtract $4 x$ from $- 2 x$ .

$x^{2} - 6 x + 1 = - 4$

Step 3.3

Add $4$ to both sides of the equation.

$x^{2} - 6 x + 1 + 4 = 0$

Step 3.4

Add $1$ and $4$ .

$x^{2} - 6 x + 5 = 0$

Step 3.5

Factor $x^{2} - 6 x + 5$ using the AC method.

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Step 3.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $5$ and whose sum is $- 6$ .

$- 5, - 1$

Step 3.5.2

Write the factored form using these integers.

$(x - 5) (x - 1) = 0$

Step 3.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 5 = 0$

$x - 1 = 0$

Step 3.7

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 3.7.2

Add $5$ to both sides of the equation.

$x = 5$

Step 3.8

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.8.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.9

The final solution is all the values that make $(x - 5) (x - 1) = 0$ true.

$x = 5, 1$