Trigonometry Examples

$\sqrt{2 a - 9} - \sqrt{a - 5} = 1$

Step 1

Add $\sqrt{a - 5}$ to both sides of the equation.

$\sqrt{2 a - 9} = 1 + \sqrt{a - 5}$

Step 2

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

${\sqrt{2 a - 9}}^{2} = {(1 + \sqrt{a - 5})}^{2}$

Step 3

Simplify each side of the equation.

Tap for more steps...

Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 a - 9}$ as ${(2 a - 9)}^{\frac{1}{2}}$ .

${({(2 a - 9)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{a - 5})}^{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${({(2 a - 9)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1

Multiply the exponents in ${({(2 a - 9)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1.1

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

${(2 a - 9)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{a - 5})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

${(2 a - 9)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{a - 5})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 a - 9)}^{1} = {(1 + \sqrt{a - 5})}^{2}$

Step 3.2.1.2

Simplify.

$2 a - 9 = {(1 + \sqrt{a - 5})}^{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify ${(1 + \sqrt{a - 5})}^{2}$ .

Tap for more steps...

Step 3.3.1.1

Rewrite ${(1 + \sqrt{a - 5})}^{2}$ as $(1 + \sqrt{a - 5}) (1 + \sqrt{a - 5})$ .

$2 a - 9 = (1 + \sqrt{a - 5}) (1 + \sqrt{a - 5})$

Step 3.3.1.2

Expand $(1 + \sqrt{a - 5}) (1 + \sqrt{a - 5})$ using the FOIL Method.

Tap for more steps...

Step 3.3.1.2.1

Apply the distributive property.

$2 a - 9 = 1 (1 + \sqrt{a - 5}) + \sqrt{a - 5} (1 + \sqrt{a - 5})$

Step 3.3.1.2.2

Apply the distributive property.

$2 a - 9 = 1 \cdot 1 + 1 \sqrt{a - 5} + \sqrt{a - 5} (1 + \sqrt{a - 5})$

Step 3.3.1.2.3

Apply the distributive property.

$2 a - 9 = 1 \cdot 1 + 1 \sqrt{a - 5} + \sqrt{a - 5} \cdot 1 + \sqrt{a - 5} \sqrt{a - 5}$

Step 3.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.3.1.1

Multiply $1$ by $1$ .

$2 a - 9 = 1 + 1 \sqrt{a - 5} + \sqrt{a - 5} \cdot 1 + \sqrt{a - 5} \sqrt{a - 5}$

Step 3.3.1.3.1.2

Multiply $\sqrt{a - 5}$ by $1$ .

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} \cdot 1 + \sqrt{a - 5} \sqrt{a - 5}$

Step 3.3.1.3.1.3

Multiply $\sqrt{a - 5}$ by $1$ .

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + \sqrt{a - 5} \sqrt{a - 5}$

Step 3.3.1.3.1.4

Multiply $\sqrt{a - 5} \sqrt{a - 5}$ .

Tap for more steps...

Step 3.3.1.3.1.4.1

Raise $\sqrt{a - 5}$ to the power of $1$ .

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + {\sqrt{a - 5}}^{1} \sqrt{a - 5}$

Step 3.3.1.3.1.4.2

Raise $\sqrt{a - 5}$ to the power of $1$ .

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + {\sqrt{a - 5}}^{1} {\sqrt{a - 5}}^{1}$

Step 3.3.1.3.1.4.3

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

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + {\sqrt{a - 5}}^{1 + 1}$

Step 3.3.1.3.1.4.4

Add $1$ and $1$ .

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + {\sqrt{a - 5}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{a - 5}}^{2}$ as $a - 5$ .

Tap for more steps...

Step 3.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a - 5}$ as ${(a - 5)}^{\frac{1}{2}}$ .

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + {({(a - 5)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

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

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + {(a - 5)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

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

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + {(a - 5)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.3.1.5.4.1

Cancel the common factor.

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + {(a - 5)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + {(a - 5)}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$2 a - 9 = 1 + \sqrt{a - 5} + \sqrt{a - 5} + a - 5$

Step 3.3.1.3.2

Subtract $5$ from $1$ .

$2 a - 9 = - 4 + \sqrt{a - 5} + \sqrt{a - 5} + a$

Step 3.3.1.3.3

Add $\sqrt{a - 5}$ and $\sqrt{a - 5}$ .

$2 a - 9 = - 4 + 2 \sqrt{a - 5} + a$

Step 4

Solve for $2 \sqrt{a - 5}$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $- 4 + 2 \sqrt{a - 5} + a = 2 a - 9$ .

$- 4 + 2 \sqrt{a - 5} + a = 2 a - 9$

Step 4.2

Move all terms not containing $2 \sqrt{a - 5}$ to the right side of the equation.

Tap for more steps...

Step 4.2.1

Add $4$ to both sides of the equation.

$2 \sqrt{a - 5} + a = 2 a - 9 + 4$

Step 4.2.2

Subtract $a$ from both sides of the equation.

$2 \sqrt{a - 5} = 2 a - 9 + 4 - a$

Step 4.2.3

Subtract $a$ from $2 a$ .

$2 \sqrt{a - 5} = a - 9 + 4$

Step 4.2.4

Add $- 9$ and $4$ .

$2 \sqrt{a - 5} = a - 5$

Step 5

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

${(2 \sqrt{a - 5})}^{2} = {(a - 5)}^{2}$

Step 6

Simplify each side of the equation.

Tap for more steps...

Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a - 5}$ as ${(a - 5)}^{\frac{1}{2}}$ .

${(2 {(a - 5)}^{\frac{1}{2}})}^{2} = {(a - 5)}^{2}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Simplify ${(2 {(a - 5)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 6.2.1.1

Apply the product rule to $2 {(a - 5)}^{\frac{1}{2}}$ .

$2^{2} {({(a - 5)}^{\frac{1}{2}})}^{2} = {(a - 5)}^{2}$

Step 6.2.1.2

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

$4 {({(a - 5)}^{\frac{1}{2}})}^{2} = {(a - 5)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(a - 5)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 6.2.1.3.1

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

$4 {(a - 5)}^{\frac{1}{2} \cdot 2} = {(a - 5)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.1.3.2.1

Cancel the common factor.

$4 {(a - 5)}^{\frac{1}{2} \cdot 2} = {(a - 5)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$4 {(a - 5)}^{1} = {(a - 5)}^{2}$

Step 6.2.1.4

Simplify.

$4 (a - 5) = {(a - 5)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$4 a + 4 \cdot - 5 = {(a - 5)}^{2}$

Step 6.2.1.6

Multiply $4$ by $- 5$ .

$4 a - 20 = {(a - 5)}^{2}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Simplify ${(a - 5)}^{2}$ .

Tap for more steps...

Step 6.3.1.1

Rewrite ${(a - 5)}^{2}$ as $(a - 5) (a - 5)$ .

$4 a - 20 = (a - 5) (a - 5)$

Step 6.3.1.2

Expand $(a - 5) (a - 5)$ using the FOIL Method.

Tap for more steps...

Step 6.3.1.2.1

Apply the distributive property.

$4 a - 20 = a (a - 5) - 5 (a - 5)$

Step 6.3.1.2.2

Apply the distributive property.

$4 a - 20 = a \cdot a + a \cdot - 5 - 5 (a - 5)$

Step 6.3.1.2.3

Apply the distributive property.

$4 a - 20 = a \cdot a + a \cdot - 5 - 5 a - 5 \cdot - 5$

Step 6.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 6.3.1.3.1

Simplify each term.

Tap for more steps...

Step 6.3.1.3.1.1

Multiply $a$ by $a$ .

$4 a - 20 = a^{2} + a \cdot - 5 - 5 a - 5 \cdot - 5$

Step 6.3.1.3.1.2

Move $- 5$ to the left of $a$ .

$4 a - 20 = a^{2} - 5 \cdot a - 5 a - 5 \cdot - 5$

Step 6.3.1.3.1.3

Multiply $- 5$ by $- 5$ .

$4 a - 20 = a^{2} - 5 a - 5 a + 25$

Step 6.3.1.3.2

Subtract $5 a$ from $- 5 a$ .

$4 a - 20 = a^{2} - 10 a + 25$

Step 7

Solve for $a$ .

Tap for more steps...

Step 7.1

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

$a^{2} - 10 a + 25 = 4 a - 20$

Step 7.2

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

Tap for more steps...

Step 7.2.1

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

$a^{2} - 10 a + 25 - 4 a = - 20$

Step 7.2.2

Subtract $4 a$ from $- 10 a$ .

$a^{2} - 14 a + 25 = - 20$

Step 7.3

Add $20$ to both sides of the equation.

$a^{2} - 14 a + 25 + 20 = 0$

Step 7.4

Add $25$ and $20$ .

$a^{2} - 14 a + 45 = 0$

Step 7.5

Factor $a^{2} - 14 a + 45$ using the AC method.

Tap for more steps...

Step 7.5.1

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 $45$ and whose sum is $- 14$ .

$- 9, - 5$

Step 7.5.2

Write the factored form using these integers.

$(a - 9) (a - 5) = 0$

Step 7.6

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

$a - 9 = 0$

$a - 5 = 0$

Step 7.7

Set $a - 9$ equal to $0$ and solve for $a$ .

Tap for more steps...

Step 7.7.1

Set $a - 9$ equal to $0$ .

$a - 9 = 0$

Step 7.7.2

Add $9$ to both sides of the equation.

$a = 9$

Step 7.8

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

Tap for more steps...

Step 7.8.1

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

$a - 5 = 0$

Step 7.8.2

Add $5$ to both sides of the equation.

$a = 5$

Step 7.9

The final solution is all the values that make $(a - 9) (a - 5) = 0$ true.

$a = 9, 5$