Solve by Factoring square root of 7+6x+ square root of 4x+5=2

$\sqrt{7 + 6 x} + \sqrt{4 x + 5} = 2$

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{7 + 6 x} + \sqrt{4 x + 5} - 2 = 0$

Step 2

Move all terms not containing $\sqrt{7 + 6 x}$ to the right side of the equation.

Subtract $\sqrt{4 x + 5}$ from both sides of the equation.

$\sqrt{7 + 6 x} - 2 = - \sqrt{4 x + 5}$

Add $2$ to both sides of the equation.

$\sqrt{7 + 6 x} = - \sqrt{4 x + 5} + 2$

Step 3

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

${\sqrt{7 + 6 x}}^{2} = {(- \sqrt{4 x + 5} + 2)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 + 6 x}$ as ${(7 + 6 x)}^{\frac{1}{2}}$ .

${({(7 + 6 x)}^{\frac{1}{2}})}^{2} = {(- \sqrt{4 x + 5} + 2)}^{2}$

Simplify the left side.

Simplify ${({(7 + 6 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(7 + 6 x)}^{\frac{1}{2}})}^{2}$ .

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

${(7 + 6 x)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{4 x + 5} + 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(7 + 6 x)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{4 x + 5} + 2)}^{2}$

Rewrite the expression.

${(7 + 6 x)}^{1} = {(- \sqrt{4 x + 5} + 2)}^{2}$

Simplify.

$7 + 6 x = {(- \sqrt{4 x + 5} + 2)}^{2}$

Simplify the right side.

Simplify ${(- \sqrt{4 x + 5} + 2)}^{2}$ .

Rewrite ${(- \sqrt{4 x + 5} + 2)}^{2}$ as $(- \sqrt{4 x + 5} + 2) (- \sqrt{4 x + 5} + 2)$ .

$7 + 6 x = (- \sqrt{4 x + 5} + 2) (- \sqrt{4 x + 5} + 2)$

Expand $(- \sqrt{4 x + 5} + 2) (- \sqrt{4 x + 5} + 2)$ using the FOIL Method.

Apply the distributive property.

$7 + 6 x = - \sqrt{4 x + 5} (- \sqrt{4 x + 5} + 2) + 2 (- \sqrt{4 x + 5} + 2)$

Apply the distributive property.

$7 + 6 x = - \sqrt{4 x + 5} (- \sqrt{4 x + 5}) - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5} + 2)$

Apply the distributive property.

$7 + 6 x = - \sqrt{4 x + 5} (- \sqrt{4 x + 5}) - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Multiply $- \sqrt{4 x + 5} (- \sqrt{4 x + 5})$ .

Multiply $- 1$ by $- 1$ .

$7 + 6 x = 1 \sqrt{4 x + 5} \sqrt{4 x + 5} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Multiply $\sqrt{4 x + 5}$ by $1$ .

$7 + 6 x = \sqrt{4 x + 5} \sqrt{4 x + 5} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Raise $\sqrt{4 x + 5}$ to the power of $1$ .

$7 + 6 x = {\sqrt{4 x + 5}}^{1} \sqrt{4 x + 5} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Raise $\sqrt{4 x + 5}$ to the power of $1$ .

$7 + 6 x = {\sqrt{4 x + 5}}^{1} {\sqrt{4 x + 5}}^{1} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

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

$7 + 6 x = {\sqrt{4 x + 5}}^{1 + 1} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Add $1$ and $1$ .

$7 + 6 x = {\sqrt{4 x + 5}}^{2} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Rewrite ${\sqrt{4 x + 5}}^{2}$ as $4 x + 5$ .

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

$7 + 6 x = {({(4 x + 5)}^{\frac{1}{2}})}^{2} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

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

$7 + 6 x = {(4 x + 5)}^{\frac{1}{2} \cdot 2} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

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

$7 + 6 x = {(4 x + 5)}^{\frac{2}{2}} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$7 + 6 x = {(4 x + 5)}^{\frac{2}{2}} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Rewrite the expression.

$7 + 6 x = {(4 x + 5)}^{1} - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Simplify.

$7 + 6 x = 4 x + 5 - \sqrt{4 x + 5} \cdot 2 + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Multiply $2$ by $- 1$ .

$7 + 6 x = 4 x + 5 - 2 \sqrt{4 x + 5} + 2 (- \sqrt{4 x + 5}) + 2 \cdot 2$

Multiply $- 1$ by $2$ .

$7 + 6 x = 4 x + 5 - 2 \sqrt{4 x + 5} - 2 \sqrt{4 x + 5} + 2 \cdot 2$

Multiply $2$ by $2$ .

$7 + 6 x = 4 x + 5 - 2 \sqrt{4 x + 5} - 2 \sqrt{4 x + 5} + 4$

Add $5$ and $4$ .

$7 + 6 x = 4 x + 9 - 2 \sqrt{4 x + 5} - 2 \sqrt{4 x + 5}$

Subtract $2 \sqrt{4 x + 5}$ from $- 2 \sqrt{4 x + 5}$ .

$7 + 6 x = 4 x + 9 - 4 \sqrt{4 x + 5}$

Step 5

Solve for $- 4 \sqrt{4 x + 5}$ .

Rewrite the equation as $4 x + 9 - 4 \sqrt{4 x + 5} = 7 + 6 x$ .

$4 x + 9 - 4 \sqrt{4 x + 5} = 7 + 6 x$

Move all terms not containing $- 4 \sqrt{4 x + 5}$ to the right side of the equation.

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

$9 - 4 \sqrt{4 x + 5} = 7 + 6 x - 4 x$

Subtract $9$ from both sides of the equation.

$- 4 \sqrt{4 x + 5} = 7 + 6 x - 4 x - 9$

Subtract $9$ from $7$ .

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

Subtract $4 x$ from $6 x$ .

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

Step 6

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

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

Step 7

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $- 4 {(4 x + 5)}^{\frac{1}{2}}$ .

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

Raise $- 4$ to the power of $2$ .

$16 {({(4 x + 5)}^{\frac{1}{2}})}^{2} = {(2 x - 2)}^{2}$

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

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

$16 {(4 x + 5)}^{\frac{1}{2} \cdot 2} = {(2 x - 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(4 x + 5)}^{\frac{1}{2} \cdot 2} = {(2 x - 2)}^{2}$

Rewrite the expression.

$16 {(4 x + 5)}^{1} = {(2 x - 2)}^{2}$

Simplify.

$16 (4 x + 5) = {(2 x - 2)}^{2}$

Apply the distributive property.

$16 (4 x) + 16 \cdot 5 = {(2 x - 2)}^{2}$

Multiply.

Multiply $4$ by $16$ .

$64 x + 16 \cdot 5 = {(2 x - 2)}^{2}$

Multiply $16$ by $5$ .

$64 x + 80 = {(2 x - 2)}^{2}$

Simplify the right side.

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

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

$64 x + 80 = (2 x - 2) (2 x - 2)$

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

Apply the distributive property.

$64 x + 80 = 2 x (2 x - 2) - 2 (2 x - 2)$

Apply the distributive property.

$64 x + 80 = 2 x (2 x) + 2 x \cdot - 2 - 2 (2 x - 2)$

Apply the distributive property.

$64 x + 80 = 2 x (2 x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$64 x + 80 = 2 \cdot 2 x \cdot x + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

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

Move $x$ .

$64 x + 80 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Multiply $x$ by $x$ .

$64 x + 80 = 2 \cdot 2 x^{2} + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Multiply $2$ by $2$ .

$64 x + 80 = 4 x^{2} + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Multiply $- 2$ by $2$ .

$64 x + 80 = 4 x^{2} - 4 x - 2 (2 x) - 2 \cdot - 2$

Multiply $2$ by $- 2$ .

$64 x + 80 = 4 x^{2} - 4 x - 4 x - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$64 x + 80 = 4 x^{2} - 4 x - 4 x + 4$

Subtract $4 x$ from $- 4 x$ .

$64 x + 80 = 4 x^{2} - 8 x + 4$

Step 8

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.

$4 x^{2} - 8 x + 4 = 64 x + 80$

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

Subtract $64 x$ from both sides of the equation.

$4 x^{2} - 8 x + 4 - 64 x = 80$

Subtract $64 x$ from $- 8 x$ .

$4 x^{2} - 72 x + 4 = 80$

Subtract $80$ from both sides of the equation.

$4 x^{2} - 72 x + 4 - 80 = 0$

Subtract $80$ from $4$ .

$4 x^{2} - 72 x - 76 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 72 x - 76$ .

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

$4 (x^{2}) - 72 x - 76 = 0$

Factor $4$ out of $- 72 x$ .

$4 (x^{2}) + 4 (- 18 x) - 76 = 0$

Factor $4$ out of $- 76$ .

$4 x^{2} + 4 (- 18 x) + 4 \cdot - 19 = 0$

Factor $4$ out of $4 x^{2} + 4 (- 18 x)$ .

$4 (x^{2} - 18 x) + 4 \cdot - 19 = 0$

Factor $4$ out of $4 (x^{2} - 18 x) + 4 \cdot - 19$ .

$4 (x^{2} - 18 x - 19) = 0$

Factor.

Factor $x^{2} - 18 x - 19$ 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 $- 19$ and whose sum is $- 18$ .

$- 19, 1$

Write the factored form using these integers.

$4 ((x - 19) (x + 1)) = 0$

Remove unnecessary parentheses.

$4 (x - 19) (x + 1) = 0$

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

$x - 19 = 0$

$x + 1 = 0$

Set $x - 19$ equal to $0$ and solve for $x$ .

Set $x - 19$ equal to $0$ .

$x - 19 = 0$

Add $19$ to both sides of the equation.

$x = 19$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The final solution is all the values that make $4 (x - 19) (x + 1) = 0$ true.

$x = 19, - 1$

Step 9

Exclude the solutions that do not make $\sqrt{7 + 6 x} + \sqrt{4 x + 5} - 2 = 0$ true.

$x = - 1$

Solve by Factoring square root of 7+6x+ square root of 4x+5=2

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