Evaluate square root of 4x+5=3+ square root of 2x-6

$\sqrt{4 x + 5} = 3 + \sqrt{2 x - 6}$

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

${\sqrt{4 x + 5}}^{2} = {(3 + \sqrt{2 x - 6})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x + 5}$ as ${(4 x + 5)}^{\frac{1}{2}}$ .

${({(4 x + 5)}^{\frac{1}{2}})}^{2} = {(3 + \sqrt{2 x - 6})}^{2}$

Simplify the left side.

Simplify ${({(4 x + 5)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 x + 5)}^{\frac{1}{2}})}^{2}$ .

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

${(4 x + 5)}^{\frac{1}{2} \cdot 2} = {(3 + \sqrt{2 x - 6})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x + 5)}^{\frac{1}{2} \cdot 2} = {(3 + \sqrt{2 x - 6})}^{2}$

Rewrite the expression.

${(4 x + 5)}^{1} = {(3 + \sqrt{2 x - 6})}^{2}$

Simplify.

$4 x + 5 = {(3 + \sqrt{2 x - 6})}^{2}$

Simplify the right side.

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

Simplify with factoring out.

Factor $2$ out of $2 x - 6$ .

Factor $2$ out of $2 x$ .

$4 x + 5 = {(3 + \sqrt{2 (x) - 6})}^{2}$

Factor $2$ out of $- 6$ .

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

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

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + \sqrt{2 (x - 3)} \cdot 3 + \sqrt{2 (x - 3)} \sqrt{2 (x - 3)}$

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

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \cdot \sqrt{2 (x - 3)} + \sqrt{2 (x - 3)} \sqrt{2 (x - 3)}$

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

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

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + {\sqrt{2 (x - 3)}}^{1} \sqrt{2 (x - 3)}$

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

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + {\sqrt{2 (x - 3)}}^{1} {\sqrt{2 (x - 3)}}^{1}$

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

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + {\sqrt{2 (x - 3)}}^{1 + 1}$

Add $1$ and $1$ .

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + {\sqrt{2 (x - 3)}}^{2}$

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

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

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + {({(2 (x - 3))}^{\frac{1}{2}})}^{2}$

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

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + {(2 (x - 3))}^{\frac{1}{2} \cdot 2}$

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

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + {(2 (x - 3))}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + {(2 (x - 3))}^{\frac{2}{2}}$

Rewrite the expression.

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + {(2 (x - 3))}^{1}$

Simplify.

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + 2 (x - 3)$

Apply the distributive property.

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + 2 x + 2 \cdot - 3$

Multiply $2$ by $- 3$ .

$4 x + 5 = 9 + 3 \sqrt{2 (x - 3)} + 3 \sqrt{2 (x - 3)} + 2 x - 6$

Subtract $6$ from $9$ .

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

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

$4 x + 5 = 3 + 6 \sqrt{2 (x - 3)} + 2 x$

Solve for $6 \sqrt{2 (x - 3)}$ .

Rewrite the equation as $3 + 6 \sqrt{2 (x - 3)} + 2 x = 4 x + 5$ .

$3 + 6 \sqrt{2 (x - 3)} + 2 x = 4 x + 5$

Move all terms not containing $6 \sqrt{2 (x - 3)}$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$6 \sqrt{2 (x - 3)} + 2 x = 4 x + 5 - 3$

Subtract $2 x$ from both sides of the equation.

$6 \sqrt{2 (x - 3)} = 4 x + 5 - 3 - 2 x$

Subtract $2 x$ from $4 x$ .

$6 \sqrt{2 (x - 3)} = 2 x + 5 - 3$

Subtract $3$ from $5$ .

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

Simplify ${(6 {(2 (x - 3))}^{\frac{1}{2}})}^{2}$ .

Apply the distributive property.

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

Simplify the expression.

Multiply $2$ by $- 3$ .

${(6 {(2 x - 6)}^{\frac{1}{2}})}^{2} = {(2 x + 2)}^{2}$

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

$6^{2} {({(2 x - 6)}^{\frac{1}{2}})}^{2} = {(2 x + 2)}^{2}$

Raise $6$ to the power of $2$ .

$36 {({(2 x - 6)}^{\frac{1}{2}})}^{2} = {(2 x + 2)}^{2}$

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$36 {(2 x - 6)}^{1} = {(2 x + 2)}^{2}$

Simplify.

$36 (2 x - 6) = {(2 x + 2)}^{2}$

Apply the distributive property.

$36 (2 x) + 36 \cdot - 6 = {(2 x + 2)}^{2}$

Multiply.

Multiply $2$ by $36$ .

$72 x + 36 \cdot - 6 = {(2 x + 2)}^{2}$

Multiply $36$ by $- 6$ .

$72 x - 216 = {(2 x + 2)}^{2}$

Simplify the right side.

Simplify ${(2 x + 2)}^{2}$ .

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

$72 x - 216 = (2 x + 2) (2 x + 2)$

Expand $(2 x + 2) (2 x + 2)$ using the FOIL Method.

Apply the distributive property.

$72 x - 216 = 2 x (2 x + 2) + 2 (2 x + 2)$

Apply the distributive property.

$72 x - 216 = 2 x (2 x) + 2 x \cdot 2 + 2 (2 x + 2)$

Apply the distributive property.

$72 x - 216 = 2 x (2 x) + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$72 x - 216 = 2 \cdot 2 x \cdot x + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

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

Move $x$ .

$72 x - 216 = 2 \cdot 2 (x \cdot x) + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Multiply $x$ by $x$ .

$72 x - 216 = 2 \cdot 2 x^{2} + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Multiply $2$ by $2$ .

$72 x - 216 = 4 x^{2} + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Multiply $2$ by $2$ .

$72 x - 216 = 4 x^{2} + 4 x + 2 (2 x) + 2 \cdot 2$

Multiply $2$ by $2$ .

$72 x - 216 = 4 x^{2} + 4 x + 4 x + 2 \cdot 2$

Multiply $2$ by $2$ .

$72 x - 216 = 4 x^{2} + 4 x + 4 x + 4$

Add $4 x$ and $4 x$ .

$72 x - 216 = 4 x^{2} + 8 x + 4$

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.

$4 x^{2} + 8 x + 4 = 72 x - 216$

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

Subtract $72 x$ from both sides of the equation.

$4 x^{2} + 8 x + 4 - 72 x = - 216$

Subtract $72 x$ from $8 x$ .

$4 x^{2} - 64 x + 4 = - 216$

Add $216$ to both sides of the equation.

$4 x^{2} - 64 x + 4 + 216 = 0$

Add $4$ and $216$ .

$4 x^{2} - 64 x + 220 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 64 x + 220$ .

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) - 64 x + 220 = 0$

Factor $4$ out of $- 64 x$ .

$4 (x^{2}) + 4 (- 16 x) + 220 = 0$

Factor $4$ out of $220$ .

$4 x^{2} + 4 (- 16 x) + 4 \cdot 55 = 0$

Factor $4$ out of $4 x^{2} + 4 (- 16 x)$ .

$4 (x^{2} - 16 x) + 4 \cdot 55 = 0$

Factor $4$ out of $4 (x^{2} - 16 x) + 4 \cdot 55$ .

$4 (x^{2} - 16 x + 55) = 0$

Factor.

Factor $x^{2} - 16 x + 55$ 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 $55$ and whose sum is $- 16$ .

$- 11, - 5$

Write the factored form using these integers.

$4 ((x - 11) (x - 5)) = 0$

Remove unnecessary parentheses.

$4 (x - 11) (x - 5) = 0$

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

$x - 11 = 0$

$x - 5 = 0$

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

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

$x - 11 = 0$

Add $11$ to both sides of the equation.

$x = 11$

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

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $4 (x - 11) (x - 5) = 0$ true.

$x = 11, 5$

Evaluate square root of 4x+5=3+ square root of 2x-6

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