Trigonometry Examples

$\sqrt{2 x - 2} = 1 + \sqrt{4 - x}$

Step 1

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

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

Expand $(1 + \sqrt{4 - x}) (1 + \sqrt{4 - x})$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Apply the distributive property.

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

Step 2.3.1.2.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 2.3.1.3.1.2

Multiply $\sqrt{4 - x}$ by $1$ .

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

Step 2.3.1.3.1.3

Multiply $\sqrt{4 - x}$ by $1$ .

$2 x - 2 = 1 + \sqrt{4 - x} + \sqrt{4 - x} + \sqrt{4 - x} \sqrt{4 - x}$

Step 2.3.1.3.1.4

Multiply $\sqrt{4 - x} \sqrt{4 - x}$ .

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Step 2.3.1.3.1.4.1

Raise $\sqrt{4 - x}$ to the power of $1$ .

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

Step 2.3.1.3.1.4.2

Raise $\sqrt{4 - x}$ to the power of $1$ .

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

Step 2.3.1.3.1.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.1.3.1.4.4

Add $1$ and $1$ .

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

Step 2.3.1.3.1.5

Rewrite ${\sqrt{4 - x}}^{2}$ as $4 - x$ .

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Step 2.3.1.3.1.5.1

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

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

Step 2.3.1.3.1.5.2

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

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

Step 2.3.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.5.4.1

Cancel the common factor.

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

Step 2.3.1.3.1.5.4.2

Rewrite the expression.

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

Step 2.3.1.3.1.5.5

Simplify.

$2 x - 2 = 1 + \sqrt{4 - x} + \sqrt{4 - x} + 4 - x$

Step 2.3.1.3.2

Add $1$ and $4$ .

$2 x - 2 = 5 + \sqrt{4 - x} + \sqrt{4 - x} - x$

Step 2.3.1.3.3

Add $\sqrt{4 - x}$ and $\sqrt{4 - x}$ .

$2 x - 2 = 5 + 2 \sqrt{4 - x} - x$

Step 3

Solve for $2 \sqrt{4 - x}$ .

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Step 3.1

Rewrite the equation as $5 + 2 \sqrt{4 - x} - x = 2 x - 2$ .

$5 + 2 \sqrt{4 - x} - x = 2 x - 2$

Step 3.2

Move all terms not containing $2 \sqrt{4 - x}$ to the right side of the equation.

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Step 3.2.1

Subtract $5$ from both sides of the equation.

$2 \sqrt{4 - x} - x = 2 x - 2 - 5$

Step 3.2.2

Add $x$ to both sides of the equation.

$2 \sqrt{4 - x} = 2 x - 2 - 5 + x$

Step 3.2.3

Add $2 x$ and $x$ .

$2 \sqrt{4 - x} = 3 x - 2 - 5$

Step 3.2.4

Subtract $5$ from $- 2$ .

$2 \sqrt{4 - x} = 3 x - 7$

Step 4

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

${(2 \sqrt{4 - x})}^{2} = {(3 x - 7)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

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

${(2 {(4 - x)}^{\frac{1}{2}})}^{2} = {(3 x - 7)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

$2^{2} {({(4 - x)}^{\frac{1}{2}})}^{2} = {(3 x - 7)}^{2}$

Step 5.2.1.2

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

$4 {({(4 - x)}^{\frac{1}{2}})}^{2} = {(3 x - 7)}^{2}$

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

$4 {(4 - x)}^{\frac{1}{2} \cdot 2} = {(3 x - 7)}^{2}$

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

$4 {(4 - x)}^{\frac{1}{2} \cdot 2} = {(3 x - 7)}^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$4 {(4 - x)}^{1} = {(3 x - 7)}^{2}$

Step 5.2.1.4

Simplify.

$4 (4 - x) = {(3 x - 7)}^{2}$

Step 5.2.1.5

Apply the distributive property.

$4 \cdot 4 + 4 (- x) = {(3 x - 7)}^{2}$

Step 5.2.1.6

Multiply.

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Step 5.2.1.6.1

Multiply $4$ by $4$ .

$16 + 4 (- x) = {(3 x - 7)}^{2}$

Step 5.2.1.6.2

Multiply $- 1$ by $4$ .

$16 - 4 x = {(3 x - 7)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(3 x - 7)}^{2}$ .

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Step 5.3.1.1

Rewrite ${(3 x - 7)}^{2}$ as $(3 x - 7) (3 x - 7)$ .

$16 - 4 x = (3 x - 7) (3 x - 7)$

Step 5.3.1.2

Expand $(3 x - 7) (3 x - 7)$ using the FOIL Method.

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Step 5.3.1.2.1

Apply the distributive property.

$16 - 4 x = 3 x (3 x - 7) - 7 (3 x - 7)$

Step 5.3.1.2.2

Apply the distributive property.

$16 - 4 x = 3 x (3 x) + 3 x \cdot - 7 - 7 (3 x - 7)$

Step 5.3.1.2.3

Apply the distributive property.

$16 - 4 x = 3 x (3 x) + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7$

Step 5.3.1.3

Simplify and combine like terms.

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Step 5.3.1.3.1

Simplify each term.

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Step 5.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$16 - 4 x = 3 \cdot 3 x \cdot x + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7$

Step 5.3.1.3.1.2

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

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Step 5.3.1.3.1.2.1

Move $x$ .

$16 - 4 x = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7$

Step 5.3.1.3.1.2.2

Multiply $x$ by $x$ .

$16 - 4 x = 3 \cdot 3 x^{2} + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7$

Step 5.3.1.3.1.3

Multiply $3$ by $3$ .

$16 - 4 x = 9 x^{2} + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7$

Step 5.3.1.3.1.4

Multiply $- 7$ by $3$ .

$16 - 4 x = 9 x^{2} - 21 x - 7 (3 x) - 7 \cdot - 7$

Step 5.3.1.3.1.5

Multiply $3$ by $- 7$ .

$16 - 4 x = 9 x^{2} - 21 x - 21 x - 7 \cdot - 7$

Step 5.3.1.3.1.6

Multiply $- 7$ by $- 7$ .

$16 - 4 x = 9 x^{2} - 21 x - 21 x + 49$

Step 5.3.1.3.2

Subtract $21 x$ from $- 21 x$ .

$16 - 4 x = 9 x^{2} - 42 x + 49$

Step 6

Solve for $x$ .

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Step 6.1

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

$9 x^{2} - 42 x + 49 = 16 - 4 x$

Step 6.2

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

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Step 6.2.1

Add $4 x$ to both sides of the equation.

$9 x^{2} - 42 x + 49 + 4 x = 16$

Step 6.2.2

Add $- 42 x$ and $4 x$ .

$9 x^{2} - 38 x + 49 = 16$

Step 6.3

Subtract $16$ from both sides of the equation.

$9 x^{2} - 38 x + 49 - 16 = 0$

Step 6.4

Subtract $16$ from $49$ .

$9 x^{2} - 38 x + 33 = 0$

Step 6.5

Factor by grouping.

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Step 6.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot 33 = 297$ and whose sum is $b = - 38$ .

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Step 6.5.1.1

Factor $- 38$ out of $- 38 x$ .

$9 x^{2} - 38 x + 33 = 0$

Step 6.5.1.2

Rewrite $- 38$ as $- 11$ plus $- 27$

$9 x^{2} + (- 11 - 27) x + 33 = 0$

Step 6.5.1.3

Apply the distributive property.

$9 x^{2} - 11 x - 27 x + 33 = 0$

Step 6.5.2

Factor out the greatest common factor from each group.

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Step 6.5.2.1

Group the first two terms and the last two terms.

$(9 x^{2} - 11 x) - 27 x + 33 = 0$

Step 6.5.2.2

Factor out the greatest common factor (GCF) from each group.

$x (9 x - 11) - 3 (9 x - 11) = 0$

Step 6.5.3

Factor the polynomial by factoring out the greatest common factor, $9 x - 11$ .

$(9 x - 11) (x - 3) = 0$

Step 6.6

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

$9 x - 11 = 0$

$x - 3 = 0$

Step 6.7

Set $9 x - 11$ equal to $0$ and solve for $x$ .

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Step 6.7.1

Set $9 x - 11$ equal to $0$ .

$9 x - 11 = 0$

Step 6.7.2

Solve $9 x - 11 = 0$ for $x$ .

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Step 6.7.2.1

Add $11$ to both sides of the equation.

$9 x = 11$

Step 6.7.2.2

Divide each term in $9 x = 11$ by $9$ and simplify.

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Step 6.7.2.2.1

Divide each term in $9 x = 11$ by $9$ .

$\frac{9 x}{9} = \frac{11}{9}$

Step 6.7.2.2.2

Simplify the left side.

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Step 6.7.2.2.2.1

Cancel the common factor of $9$ .

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Step 6.7.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{11}{9}$

Step 6.7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{9}$

Step 6.8

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 6.8.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 6.8.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6.9

The final solution is all the values that make $(9 x - 11) (x - 3) = 0$ true.

$x = \frac{11}{9}, 3$

Step 7

Exclude the solutions that do not make $\sqrt{2 x - 2} = 1 + \sqrt{4 - x}$ true.

$x = 3$