Trigonometry Examples

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

Step 1

Subtract $x$ from both sides of the equation.

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

Step 2

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

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

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the product rule to $- {(4 x - 4)}^{\frac{1}{2}}$ .

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

Step 3.2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 3.2.1.3

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

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

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

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

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

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

Step 3.2.1.4.2.2

Rewrite the expression.

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

Step 3.2.1.5

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite ${(4 - x)}^{2}$ as $(4 - x) (4 - x)$ .

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

Step 3.3.1.2

Expand $(4 - x) (4 - x)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

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

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $4$ by $4$ .

$4 x - 4 = 16 + 4 (- x) - x \cdot 4 - x (- x)$

Step 3.3.1.3.1.2

Multiply $- 1$ by $4$ .

$4 x - 4 = 16 - 4 x - x \cdot 4 - x (- x)$

Step 3.3.1.3.1.3

Multiply $4$ by $- 1$ .

$4 x - 4 = 16 - 4 x - 4 x - x (- x)$

Step 3.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$4 x - 4 = 16 - 4 x - 4 x - 1 \cdot - 1 x \cdot x$

Step 3.3.1.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.3.1.5.1

Move $x$ .

$4 x - 4 = 16 - 4 x - 4 x - 1 \cdot - 1 (x \cdot x)$

Step 3.3.1.3.1.5.2

Multiply $x$ by $x$ .

$4 x - 4 = 16 - 4 x - 4 x - 1 \cdot - 1 x^{2}$

Step 3.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$4 x - 4 = 16 - 4 x - 4 x + 1 x^{2}$

Step 3.3.1.3.1.7

Multiply $x^{2}$ by $1$ .

$4 x - 4 = 16 - 4 x - 4 x + x^{2}$

Step 3.3.1.3.2

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

$4 x - 4 = 16 - 8 x + x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$16 - 8 x + x^{2} = 4 x - 4$

Step 4.2

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

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Step 4.2.1

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

$16 - 8 x + x^{2} - 4 x = - 4$

Step 4.2.2

Subtract $4 x$ from $- 8 x$ .

$16 + x^{2} - 12 x = - 4$

Step 4.3

Add $4$ to both sides of the equation.

$16 + x^{2} - 12 x + 4 = 0$

Step 4.4

Add $16$ and $4$ .

$x^{2} - 12 x + 20 = 0$

Step 4.5

Factor $x^{2} - 12 x + 20$ using the AC method.

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Step 4.5.1

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 $20$ and whose sum is $- 12$ .

$- 10, - 2$

Step 4.5.2

Write the factored form using these integers.

$(x - 10) (x - 2) = 0$

Step 4.6

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

$x - 10 = 0$

$x - 2 = 0$

Step 4.7

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 4.7.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 4.7.2

Add $10$ to both sides of the equation.

$x = 10$

Step 4.8

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 4.8.1

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

$x - 2 = 0$

Step 4.8.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.9

The final solution is all the values that make $(x - 10) (x - 2) = 0$ true.

$x = 10, 2$

Step 5

Exclude the solutions that do not make $x - \sqrt{4 x - 4} = 4$ true.

$x = 10$