Solve for x square root of 7x+2- square root of 3x+6=6

$\sqrt{7 x + 2} - \sqrt{3 x + 6} = 6$

Step 1

Solve for $\sqrt{7 x + 2}$ .

Factor $3$ out of $3 x + 6$ .

Factor $3$ out of $3 x$ .

$\sqrt{7 x + 2} - \sqrt{3 (x) + 6} = 6$

Factor $3$ out of $6$ .

$\sqrt{7 x + 2} - \sqrt{3 x + 3 \cdot 2} = 6$

Factor $3$ out of $3 x + 3 \cdot 2$ .

$\sqrt{7 x + 2} - \sqrt{3 (x + 2)} = 6$

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

$\sqrt{7 x + 2} = 6 + \sqrt{3 (x + 2)}$

Step 2

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

${\sqrt{7 x + 2}}^{2} = {(6 + \sqrt{3 (x + 2)})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x + 2}$ as ${(7 x + 2)}^{\frac{1}{2}}$ .

${({(7 x + 2)}^{\frac{1}{2}})}^{2} = {(6 + \sqrt{3 (x + 2)})}^{2}$

Simplify the left side.

Simplify ${({(7 x + 2)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(7 x + 2)}^{\frac{1}{2}})}^{2}$ .

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

${(7 x + 2)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{3 (x + 2)})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(7 x + 2)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{3 (x + 2)})}^{2}$

Rewrite the expression.

${(7 x + 2)}^{1} = {(6 + \sqrt{3 (x + 2)})}^{2}$

Simplify.

$7 x + 2 = {(6 + \sqrt{3 (x + 2)})}^{2}$

Simplify the right side.

Simplify ${(6 + \sqrt{3 (x + 2)})}^{2}$ .

Rewrite ${(6 + \sqrt{3 (x + 2)})}^{2}$ as $(6 + \sqrt{3 (x + 2)}) (6 + \sqrt{3 (x + 2)})$ .

$7 x + 2 = (6 + \sqrt{3 (x + 2)}) (6 + \sqrt{3 (x + 2)})$

Expand $(6 + \sqrt{3 (x + 2)}) (6 + \sqrt{3 (x + 2)})$ using the FOIL Method.

Apply the distributive property.

$7 x + 2 = 6 (6 + \sqrt{3 (x + 2)}) + \sqrt{3 (x + 2)} (6 + \sqrt{3 (x + 2)})$

Apply the distributive property.

$7 x + 2 = 6 \cdot 6 + 6 \sqrt{3 (x + 2)} + \sqrt{3 (x + 2)} (6 + \sqrt{3 (x + 2)})$

Apply the distributive property.

$7 x + 2 = 6 \cdot 6 + 6 \sqrt{3 (x + 2)} + \sqrt{3 (x + 2)} \cdot 6 + \sqrt{3 (x + 2)} \sqrt{3 (x + 2)}$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + \sqrt{3 (x + 2)} \cdot 6 + \sqrt{3 (x + 2)} \sqrt{3 (x + 2)}$

Move $6$ to the left of $\sqrt{3 (x + 2)}$ .

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \cdot \sqrt{3 (x + 2)} + \sqrt{3 (x + 2)} \sqrt{3 (x + 2)}$

Multiply $\sqrt{3 (x + 2)} \sqrt{3 (x + 2)}$ .

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

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + {\sqrt{3 (x + 2)}}^{1} \sqrt{3 (x + 2)}$

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

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + {\sqrt{3 (x + 2)}}^{1} {\sqrt{3 (x + 2)}}^{1}$

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

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + {\sqrt{3 (x + 2)}}^{1 + 1}$

Add $1$ and $1$ .

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + {\sqrt{3 (x + 2)}}^{2}$

Rewrite ${\sqrt{3 (x + 2)}}^{2}$ as $3 (x + 2)$ .

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

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + {({(3 (x + 2))}^{\frac{1}{2}})}^{2}$

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

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + {(3 (x + 2))}^{\frac{1}{2} \cdot 2}$

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

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + {(3 (x + 2))}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + {(3 (x + 2))}^{\frac{2}{2}}$

Rewrite the expression.

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + {(3 (x + 2))}^{1}$

Simplify.

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + 3 (x + 2)$

Apply the distributive property.

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + 3 x + 3 \cdot 2$

Multiply $3$ by $2$ .

$7 x + 2 = 36 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + 3 x + 6$

Add $36$ and $6$ .

$7 x + 2 = 42 + 6 \sqrt{3 (x + 2)} + 6 \sqrt{3 (x + 2)} + 3 x$

Add $6 \sqrt{3 (x + 2)}$ and $6 \sqrt{3 (x + 2)}$ .

$7 x + 2 = 42 + 12 \sqrt{3 (x + 2)} + 3 x$

Step 4

Solve for $12 \sqrt{3 (x + 2)}$ .

Rewrite the equation as $42 + 12 \sqrt{3 (x + 2)} + 3 x = 7 x + 2$ .

$42 + 12 \sqrt{3 (x + 2)} + 3 x = 7 x + 2$

Move all terms not containing $12 \sqrt{3 (x + 2)}$ to the right side of the equation.

Subtract $42$ from both sides of the equation.

$12 \sqrt{3 (x + 2)} + 3 x = 7 x + 2 - 42$

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

$12 \sqrt{3 (x + 2)} = 7 x + 2 - 42 - 3 x$

Subtract $3 x$ from $7 x$ .

$12 \sqrt{3 (x + 2)} = 4 x + 2 - 42$

Subtract $42$ from $2$ .

$12 \sqrt{3 (x + 2)} = 4 x - 40$

Step 5

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

${(12 \sqrt{3 (x + 2)})}^{2} = {(4 x - 40)}^{2}$

Step 6

Simplify each side of the equation.

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

${(12 {(3 (x + 2))}^{\frac{1}{2}})}^{2} = {(4 x - 40)}^{2}$

Simplify the left side.

Simplify ${(12 {(3 (x + 2))}^{\frac{1}{2}})}^{2}$ .

Apply the distributive property.

${(12 {(3 x + 3 \cdot 2)}^{\frac{1}{2}})}^{2} = {(4 x - 40)}^{2}$

Simplify the expression.

Multiply $3$ by $2$ .

${(12 {(3 x + 6)}^{\frac{1}{2}})}^{2} = {(4 x - 40)}^{2}$

Apply the product rule to $12 {(3 x + 6)}^{\frac{1}{2}}$ .

$12^{2} {({(3 x + 6)}^{\frac{1}{2}})}^{2} = {(4 x - 40)}^{2}$

Raise $12$ to the power of $2$ .

$144 {({(3 x + 6)}^{\frac{1}{2}})}^{2} = {(4 x - 40)}^{2}$

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

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

$144 {(3 x + 6)}^{\frac{1}{2} \cdot 2} = {(4 x - 40)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$144 {(3 x + 6)}^{\frac{1}{2} \cdot 2} = {(4 x - 40)}^{2}$

Rewrite the expression.

$144 {(3 x + 6)}^{1} = {(4 x - 40)}^{2}$

Simplify.

$144 (3 x + 6) = {(4 x - 40)}^{2}$

Apply the distributive property.

$144 (3 x) + 144 \cdot 6 = {(4 x - 40)}^{2}$

Multiply.

Multiply $3$ by $144$ .

$432 x + 144 \cdot 6 = {(4 x - 40)}^{2}$

Multiply $144$ by $6$ .

$432 x + 864 = {(4 x - 40)}^{2}$

Simplify the right side.

Simplify ${(4 x - 40)}^{2}$ .

Rewrite ${(4 x - 40)}^{2}$ as $(4 x - 40) (4 x - 40)$ .

$432 x + 864 = (4 x - 40) (4 x - 40)$

Expand $(4 x - 40) (4 x - 40)$ using the FOIL Method.

Apply the distributive property.

$432 x + 864 = 4 x (4 x - 40) - 40 (4 x - 40)$

Apply the distributive property.

$432 x + 864 = 4 x (4 x) + 4 x \cdot - 40 - 40 (4 x - 40)$

Apply the distributive property.

$432 x + 864 = 4 x (4 x) + 4 x \cdot - 40 - 40 (4 x) - 40 \cdot - 40$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$432 x + 864 = 4 \cdot 4 x \cdot x + 4 x \cdot - 40 - 40 (4 x) - 40 \cdot - 40$

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

Move $x$ .

$432 x + 864 = 4 \cdot 4 (x \cdot x) + 4 x \cdot - 40 - 40 (4 x) - 40 \cdot - 40$

Multiply $x$ by $x$ .

$432 x + 864 = 4 \cdot 4 x^{2} + 4 x \cdot - 40 - 40 (4 x) - 40 \cdot - 40$

Multiply $4$ by $4$ .

$432 x + 864 = 16 x^{2} + 4 x \cdot - 40 - 40 (4 x) - 40 \cdot - 40$

Multiply $- 40$ by $4$ .

$432 x + 864 = 16 x^{2} - 160 x - 40 (4 x) - 40 \cdot - 40$

Multiply $4$ by $- 40$ .

$432 x + 864 = 16 x^{2} - 160 x - 160 x - 40 \cdot - 40$

Multiply $- 40$ by $- 40$ .

$432 x + 864 = 16 x^{2} - 160 x - 160 x + 1600$

Subtract $160 x$ from $- 160 x$ .

$432 x + 864 = 16 x^{2} - 320 x + 1600$

Step 7

Solve for $x$ .

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

$16 x^{2} - 320 x + 1600 = 432 x + 864$

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

Subtract $432 x$ from both sides of the equation.

$16 x^{2} - 320 x + 1600 - 432 x = 864$

Subtract $432 x$ from $- 320 x$ .

$16 x^{2} - 752 x + 1600 = 864$

Subtract $864$ from both sides of the equation.

$16 x^{2} - 752 x + 1600 - 864 = 0$

Subtract $864$ from $1600$ .

$16 x^{2} - 752 x + 736 = 0$

Factor the left side of the equation.

Factor $16$ out of $16 x^{2} - 752 x + 736$ .

Factor $16$ out of $16 x^{2}$ .

$16 (x^{2}) - 752 x + 736 = 0$

Factor $16$ out of $- 752 x$ .

$16 (x^{2}) + 16 (- 47 x) + 736 = 0$

Factor $16$ out of $736$ .

$16 x^{2} + 16 (- 47 x) + 16 \cdot 46 = 0$

Factor $16$ out of $16 x^{2} + 16 (- 47 x)$ .

$16 (x^{2} - 47 x) + 16 \cdot 46 = 0$

Factor $16$ out of $16 (x^{2} - 47 x) + 16 \cdot 46$ .

$16 (x^{2} - 47 x + 46) = 0$

Factor.

Factor $x^{2} - 47 x + 46$ using the AC method.

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 $46$ and whose sum is $- 47$ .

$- 46, - 1$

Write the factored form using these integers.

$16 ((x - 46) (x - 1)) = 0$

Remove unnecessary parentheses.

$16 (x - 46) (x - 1) = 0$

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

$x - 46 = 0$

$x - 1 = 0$

Set $x - 46$ equal to $0$ and solve for $x$ .

Set $x - 46$ equal to $0$ .

$x - 46 = 0$

Add $46$ to both sides of the equation.

$x = 46$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $16 (x - 46) (x - 1) = 0$ true.

$x = 46, 1$

Step 8

Exclude the solutions that do not make $\sqrt{7 x + 2} - \sqrt{3 x + 6} = 6$ true.

$x = 46$

Solve for x square root of 7x+2- square root of 3x+6=6

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