Solve for c 2 square root of 21-c- square root of 3c-6=5

$2 \sqrt{21 - c} - \sqrt{3 c - 6} = 5$

Step 1

Solve for $2 \sqrt{21 - c}$ .

Factor $3$ out of $3 c - 6$ .

Factor $3$ out of $3 c$ .

$2 \sqrt{21 - c} - \sqrt{3 (c) - 6} = 5$

Factor $3$ out of $- 6$ .

$2 \sqrt{21 - c} - \sqrt{3 c + 3 \cdot - 2} = 5$

Factor $3$ out of $3 c + 3 \cdot - 2$ .

$2 \sqrt{21 - c} - \sqrt{3 (c - 2)} = 5$

Add $\sqrt{3 (c - 2)}$ to both sides of the equation.

$2 \sqrt{21 - c} = 5 + \sqrt{3 (c - 2)}$

Step 2

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

${(2 \sqrt{21 - c})}^{2} = {(5 + \sqrt{3 (c - 2)})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21 - c}$ as ${(21 - c)}^{\frac{1}{2}}$ .

${(2 {(21 - c)}^{\frac{1}{2}})}^{2} = {(5 + \sqrt{3 (c - 2)})}^{2}$

Simplify the left side.

Simplify ${(2 {(21 - c)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 {(21 - c)}^{\frac{1}{2}}$ .

$2^{2} {({(21 - c)}^{\frac{1}{2}})}^{2} = {(5 + \sqrt{3 (c - 2)})}^{2}$

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

$4 {({(21 - c)}^{\frac{1}{2}})}^{2} = {(5 + \sqrt{3 (c - 2)})}^{2}$

Multiply the exponents in ${({(21 - c)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(21 - c)}^{\frac{1}{2} \cdot 2} = {(5 + \sqrt{3 (c - 2)})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(21 - c)}^{\frac{1}{2} \cdot 2} = {(5 + \sqrt{3 (c - 2)})}^{2}$

Rewrite the expression.

$4 {(21 - c)}^{1} = {(5 + \sqrt{3 (c - 2)})}^{2}$

Simplify.

$4 (21 - c) = {(5 + \sqrt{3 (c - 2)})}^{2}$

Apply the distributive property.

$4 \cdot 21 + 4 (- c) = {(5 + \sqrt{3 (c - 2)})}^{2}$

Multiply.

Multiply $4$ by $21$ .

$84 + 4 (- c) = {(5 + \sqrt{3 (c - 2)})}^{2}$

Multiply $- 1$ by $4$ .

$84 - 4 c = {(5 + \sqrt{3 (c - 2)})}^{2}$

Simplify the right side.

Simplify ${(5 + \sqrt{3 (c - 2)})}^{2}$ .

Rewrite ${(5 + \sqrt{3 (c - 2)})}^{2}$ as $(5 + \sqrt{3 (c - 2)}) (5 + \sqrt{3 (c - 2)})$ .

$84 - 4 c = (5 + \sqrt{3 (c - 2)}) (5 + \sqrt{3 (c - 2)})$

Expand $(5 + \sqrt{3 (c - 2)}) (5 + \sqrt{3 (c - 2)})$ using the FOIL Method.

Apply the distributive property.

$84 - 4 c = 5 (5 + \sqrt{3 (c - 2)}) + \sqrt{3 (c - 2)} (5 + \sqrt{3 (c - 2)})$

Apply the distributive property.

$84 - 4 c = 5 \cdot 5 + 5 \sqrt{3 (c - 2)} + \sqrt{3 (c - 2)} (5 + \sqrt{3 (c - 2)})$

Apply the distributive property.

$84 - 4 c = 5 \cdot 5 + 5 \sqrt{3 (c - 2)} + \sqrt{3 (c - 2)} \cdot 5 + \sqrt{3 (c - 2)} \sqrt{3 (c - 2)}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + \sqrt{3 (c - 2)} \cdot 5 + \sqrt{3 (c - 2)} \sqrt{3 (c - 2)}$

Move $5$ to the left of $\sqrt{3 (c - 2)}$ .

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \cdot \sqrt{3 (c - 2)} + \sqrt{3 (c - 2)} \sqrt{3 (c - 2)}$

Multiply $\sqrt{3 (c - 2)} \sqrt{3 (c - 2)}$ .

Raise $\sqrt{3 (c - 2)}$ to the power of $1$ .

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + {\sqrt{3 (c - 2)}}^{1} \sqrt{3 (c - 2)}$

Raise $\sqrt{3 (c - 2)}$ to the power of $1$ .

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + {\sqrt{3 (c - 2)}}^{1} {\sqrt{3 (c - 2)}}^{1}$

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

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + {\sqrt{3 (c - 2)}}^{1 + 1}$

Add $1$ and $1$ .

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + {\sqrt{3 (c - 2)}}^{2}$

Rewrite ${\sqrt{3 (c - 2)}}^{2}$ as $3 (c - 2)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (c - 2)}$ as ${(3 (c - 2))}^{\frac{1}{2}}$ .

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + {({(3 (c - 2))}^{\frac{1}{2}})}^{2}$

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

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + {(3 (c - 2))}^{\frac{1}{2} \cdot 2}$

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

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + {(3 (c - 2))}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + {(3 (c - 2))}^{\frac{2}{2}}$

Rewrite the expression.

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + {(3 (c - 2))}^{1}$

Simplify.

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + 3 (c - 2)$

Apply the distributive property.

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + 3 c + 3 \cdot - 2$

Multiply $3$ by $- 2$ .

$84 - 4 c = 25 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + 3 c - 6$

Subtract $6$ from $25$ .

$84 - 4 c = 19 + 5 \sqrt{3 (c - 2)} + 5 \sqrt{3 (c - 2)} + 3 c$

Add $5 \sqrt{3 (c - 2)}$ and $5 \sqrt{3 (c - 2)}$ .

$84 - 4 c = 19 + 10 \sqrt{3 (c - 2)} + 3 c$

Step 4

Solve for $10 \sqrt{3 (c - 2)}$ .

Rewrite the equation as $19 + 10 \sqrt{3 (c - 2)} + 3 c = 84 - 4 c$ .

$19 + 10 \sqrt{3 (c - 2)} + 3 c = 84 - 4 c$

Move all terms not containing $10 \sqrt{3 (c - 2)}$ to the right side of the equation.

Subtract $19$ from both sides of the equation.

$10 \sqrt{3 (c - 2)} + 3 c = 84 - 4 c - 19$

Subtract $3 c$ from both sides of the equation.

$10 \sqrt{3 (c - 2)} = 84 - 4 c - 19 - 3 c$

Subtract $19$ from $84$ .

$10 \sqrt{3 (c - 2)} = - 4 c + 65 - 3 c$

Subtract $3 c$ from $- 4 c$ .

$10 \sqrt{3 (c - 2)} = - 7 c + 65$

Step 5

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

${(10 \sqrt{3 (c - 2)})}^{2} = {(- 7 c + 65)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (c - 2)}$ as ${(3 (c - 2))}^{\frac{1}{2}}$ .

${(10 {(3 (c - 2))}^{\frac{1}{2}})}^{2} = {(- 7 c + 65)}^{2}$

Simplify the left side.

Simplify ${(10 {(3 (c - 2))}^{\frac{1}{2}})}^{2}$ .

Apply the distributive property.

${(10 {(3 c + 3 \cdot - 2)}^{\frac{1}{2}})}^{2} = {(- 7 c + 65)}^{2}$

Simplify the expression.

Multiply $3$ by $- 2$ .

${(10 {(3 c - 6)}^{\frac{1}{2}})}^{2} = {(- 7 c + 65)}^{2}$

Apply the product rule to $10 {(3 c - 6)}^{\frac{1}{2}}$ .

$10^{2} {({(3 c - 6)}^{\frac{1}{2}})}^{2} = {(- 7 c + 65)}^{2}$

Raise $10$ to the power of $2$ .

$100 {({(3 c - 6)}^{\frac{1}{2}})}^{2} = {(- 7 c + 65)}^{2}$

Multiply the exponents in ${({(3 c - 6)}^{\frac{1}{2}})}^{2}$ .

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

$100 {(3 c - 6)}^{\frac{1}{2} \cdot 2} = {(- 7 c + 65)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 {(3 c - 6)}^{\frac{1}{2} \cdot 2} = {(- 7 c + 65)}^{2}$

Rewrite the expression.

$100 {(3 c - 6)}^{1} = {(- 7 c + 65)}^{2}$

Simplify.

$100 (3 c - 6) = {(- 7 c + 65)}^{2}$

Apply the distributive property.

$100 (3 c) + 100 \cdot - 6 = {(- 7 c + 65)}^{2}$

Multiply.

Multiply $3$ by $100$ .

$300 c + 100 \cdot - 6 = {(- 7 c + 65)}^{2}$

Multiply $100$ by $- 6$ .

$300 c - 600 = {(- 7 c + 65)}^{2}$

Simplify the right side.

Simplify ${(- 7 c + 65)}^{2}$ .

Rewrite ${(- 7 c + 65)}^{2}$ as $(- 7 c + 65) (- 7 c + 65)$ .

$300 c - 600 = (- 7 c + 65) (- 7 c + 65)$

Expand $(- 7 c + 65) (- 7 c + 65)$ using the FOIL Method.

Apply the distributive property.

$300 c - 600 = - 7 c (- 7 c + 65) + 65 (- 7 c + 65)$

Apply the distributive property.

$300 c - 600 = - 7 c (- 7 c) - 7 c \cdot 65 + 65 (- 7 c + 65)$

Apply the distributive property.

$300 c - 600 = - 7 c (- 7 c) - 7 c \cdot 65 + 65 (- 7 c) + 65 \cdot 65$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$300 c - 600 = - 7 \cdot - 7 c \cdot c - 7 c \cdot 65 + 65 (- 7 c) + 65 \cdot 65$

Multiply $c$ by $c$ by adding the exponents.

Move $c$ .

$300 c - 600 = - 7 \cdot - 7 (c \cdot c) - 7 c \cdot 65 + 65 (- 7 c) + 65 \cdot 65$

Multiply $c$ by $c$ .

$300 c - 600 = - 7 \cdot - 7 c^{2} - 7 c \cdot 65 + 65 (- 7 c) + 65 \cdot 65$

Multiply $- 7$ by $- 7$ .

$300 c - 600 = 49 c^{2} - 7 c \cdot 65 + 65 (- 7 c) + 65 \cdot 65$

Multiply $65$ by $- 7$ .

$300 c - 600 = 49 c^{2} - 455 c + 65 (- 7 c) + 65 \cdot 65$

Multiply $- 7$ by $65$ .

$300 c - 600 = 49 c^{2} - 455 c - 455 c + 65 \cdot 65$

Multiply $65$ by $65$ .

$300 c - 600 = 49 c^{2} - 455 c - 455 c + 4225$

Subtract $455 c$ from $- 455 c$ .

$300 c - 600 = 49 c^{2} - 910 c + 4225$

Step 7

Solve for $c$ .

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

$49 c^{2} - 910 c + 4225 = 300 c - 600$

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

Subtract $300 c$ from both sides of the equation.

$49 c^{2} - 910 c + 4225 - 300 c = - 600$

Subtract $300 c$ from $- 910 c$ .

$49 c^{2} - 1210 c + 4225 = - 600$

Add $600$ to both sides of the equation.

$49 c^{2} - 1210 c + 4225 + 600 = 0$

Add $4225$ and $600$ .

$49 c^{2} - 1210 c + 4825 = 0$

Factor by grouping.

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 $a \cdot c = 49 \cdot 4825 = 236425$ and whose sum is $b = - 1210$ .

Factor $- 1210$ out of $- 1210 c$ .

$49 c^{2} - 1210 c + 4825 = 0$

Rewrite $- 1210$ as $- 245$ plus $- 965$

$49 c^{2} + (- 245 - 965) c + 4825 = 0$

Apply the distributive property.

$49 c^{2} - 245 c - 965 c + 4825 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(49 c^{2} - 245 c) - 965 c + 4825 = 0$

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

$49 c (c - 5) - 965 (c - 5) = 0$

Factor the polynomial by factoring out the greatest common factor, $c - 5$ .

$(c - 5) (49 c - 965) = 0$

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

$c - 5 = 0$

$49 c - 965 = 0$

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

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

$c - 5 = 0$

Add $5$ to both sides of the equation.

$c = 5$

Set $49 c - 965$ equal to $0$ and solve for $c$ .

Set $49 c - 965$ equal to $0$ .

$49 c - 965 = 0$

Solve $49 c - 965 = 0$ for $c$ .

Add $965$ to both sides of the equation.

$49 c = 965$

Divide each term in $49 c = 965$ by $49$ and simplify.

Divide each term in $49 c = 965$ by $49$ .

$\frac{49 c}{49} = \frac{965}{49}$

Simplify the left side.

Cancel the common factor of $49$ .

Cancel the common factor.

$\frac{49 c}{49} = \frac{965}{49}$

Divide $c$ by $1$ .

$c = \frac{965}{49}$

The final solution is all the values that make $(c - 5) (49 c - 965) = 0$ true.

$c = 5, \frac{965}{49}$

Step 8

Exclude the solutions that do not make $2 \sqrt{21 - c} - \sqrt{3 c - 6} = 5$ true.

$c = 5$

Solve for c 2 square root of 21-c- square root of 3c-6=5

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