Solve for x square root of x+4 = square root of 7x+2

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

Step 1

Subtract $4$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(\sqrt{7 x + 2} - 4)}^{2}$ .

Rewrite ${(\sqrt{7 x + 2} - 4)}^{2}$ as $(\sqrt{7 x + 2} - 4) (\sqrt{7 x + 2} - 4)$ .

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

Expand $(\sqrt{7 x + 2} - 4) (\sqrt{7 x + 2} - 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{7 x + 2} \sqrt{7 x + 2}$ .

Raise $\sqrt{7 x + 2}$ to the power of $1$ .

$x = {\sqrt{7 x + 2}}^{1} \sqrt{7 x + 2} + \sqrt{7 x + 2} \cdot - 4 - 4 \sqrt{7 x + 2} - 4 \cdot - 4$

Raise $\sqrt{7 x + 2}$ to the power of $1$ .

$x = {\sqrt{7 x + 2}}^{1} {\sqrt{7 x + 2}}^{1} + \sqrt{7 x + 2} \cdot - 4 - 4 \sqrt{7 x + 2} - 4 \cdot - 4$

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

$x = {\sqrt{7 x + 2}}^{1 + 1} + \sqrt{7 x + 2} \cdot - 4 - 4 \sqrt{7 x + 2} - 4 \cdot - 4$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{7 x + 2}}^{2}$ as $7 x + 2$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Move $- 4$ to the left of $\sqrt{7 x + 2}$ .

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

Multiply $- 4$ by $- 4$ .

$x = 7 x + 2 - 4 \sqrt{7 x + 2} - 4 \sqrt{7 x + 2} + 16$

Add $2$ and $16$ .

$x = 7 x + 18 - 4 \sqrt{7 x + 2} - 4 \sqrt{7 x + 2}$

Subtract $4 \sqrt{7 x + 2}$ from $- 4 \sqrt{7 x + 2}$ .

$x = 7 x + 18 - 8 \sqrt{7 x + 2}$

Step 4

Solve for $- 8 \sqrt{7 x + 2}$ .

Rewrite the equation as $7 x + 18 - 8 \sqrt{7 x + 2} = x$ .

$7 x + 18 - 8 \sqrt{7 x + 2} = x$

Move all terms not containing $- 8 \sqrt{7 x + 2}$ to the right side of the equation.

Subtract $7 x$ from both sides of the equation.

$18 - 8 \sqrt{7 x + 2} = x - 7 x$

Subtract $18$ from both sides of the equation.

$- 8 \sqrt{7 x + 2} = x - 7 x - 18$

Subtract $7 x$ from $x$ .

$- 8 \sqrt{7 x + 2} = - 6 x - 18$

Step 5

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

${(- 8 \sqrt{7 x + 2})}^{2} = {(- 6 x - 18)}^{2}$

Step 6

Simplify each side of the equation.

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

${(- 8 {(7 x + 2)}^{\frac{1}{2}})}^{2} = {(- 6 x - 18)}^{2}$

Simplify the left side.

Simplify ${(- 8 {(7 x + 2)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 8 {(7 x + 2)}^{\frac{1}{2}}$ .

${(- 8)}^{2} {({(7 x + 2)}^{\frac{1}{2}})}^{2} = {(- 6 x - 18)}^{2}$

Raise $- 8$ to the power of $2$ .

$64 {({(7 x + 2)}^{\frac{1}{2}})}^{2} = {(- 6 x - 18)}^{2}$

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

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

$64 {(7 x + 2)}^{\frac{1}{2} \cdot 2} = {(- 6 x - 18)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$64 {(7 x + 2)}^{\frac{1}{2} \cdot 2} = {(- 6 x - 18)}^{2}$

Rewrite the expression.

$64 {(7 x + 2)}^{1} = {(- 6 x - 18)}^{2}$

Simplify.

$64 (7 x + 2) = {(- 6 x - 18)}^{2}$

Apply the distributive property.

$64 (7 x) + 64 \cdot 2 = {(- 6 x - 18)}^{2}$

Multiply.

Multiply $7$ by $64$ .

$448 x + 64 \cdot 2 = {(- 6 x - 18)}^{2}$

Multiply $64$ by $2$ .

$448 x + 128 = {(- 6 x - 18)}^{2}$

Simplify the right side.

Simplify ${(- 6 x - 18)}^{2}$ .

Rewrite ${(- 6 x - 18)}^{2}$ as $(- 6 x - 18) (- 6 x - 18)$ .

$448 x + 128 = (- 6 x - 18) (- 6 x - 18)$

Expand $(- 6 x - 18) (- 6 x - 18)$ using the FOIL Method.

Apply the distributive property.

$448 x + 128 = - 6 x (- 6 x - 18) - 18 (- 6 x - 18)$

Apply the distributive property.

$448 x + 128 = - 6 x (- 6 x) - 6 x \cdot - 18 - 18 (- 6 x - 18)$

Apply the distributive property.

$448 x + 128 = - 6 x (- 6 x) - 6 x \cdot - 18 - 18 (- 6 x) - 18 \cdot - 18$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$448 x + 128 = - 6 \cdot - 6 x \cdot x - 6 x \cdot - 18 - 18 (- 6 x) - 18 \cdot - 18$

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

Move $x$ .

$448 x + 128 = - 6 \cdot - 6 (x \cdot x) - 6 x \cdot - 18 - 18 (- 6 x) - 18 \cdot - 18$

Multiply $x$ by $x$ .

$448 x + 128 = - 6 \cdot - 6 x^{2} - 6 x \cdot - 18 - 18 (- 6 x) - 18 \cdot - 18$

Multiply $- 6$ by $- 6$ .

$448 x + 128 = 36 x^{2} - 6 x \cdot - 18 - 18 (- 6 x) - 18 \cdot - 18$

Multiply $- 18$ by $- 6$ .

$448 x + 128 = 36 x^{2} + 108 x - 18 (- 6 x) - 18 \cdot - 18$

Multiply $- 6$ by $- 18$ .

$448 x + 128 = 36 x^{2} + 108 x + 108 x - 18 \cdot - 18$

Multiply $- 18$ by $- 18$ .

$448 x + 128 = 36 x^{2} + 108 x + 108 x + 324$

Add $108 x$ and $108 x$ .

$448 x + 128 = 36 x^{2} + 216 x + 324$

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.

$36 x^{2} + 216 x + 324 = 448 x + 128$

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

Subtract $448 x$ from both sides of the equation.

$36 x^{2} + 216 x + 324 - 448 x = 128$

Subtract $448 x$ from $216 x$ .

$36 x^{2} - 232 x + 324 = 128$

Subtract $128$ from both sides of the equation.

$36 x^{2} - 232 x + 324 - 128 = 0$

Subtract $128$ from $324$ .

$36 x^{2} - 232 x + 196 = 0$

Factor the left side of the equation.

Factor $4$ out of $36 x^{2} - 232 x + 196$ .

Factor $4$ out of $36 x^{2}$ .

$4 (9 x^{2}) - 232 x + 196 = 0$

Factor $4$ out of $- 232 x$ .

$4 (9 x^{2}) + 4 (- 58 x) + 196 = 0$

Factor $4$ out of $196$ .

$4 (9 x^{2}) + 4 (- 58 x) + 4 (49) = 0$

Factor $4$ out of $4 (9 x^{2}) + 4 (- 58 x)$ .

$4 (9 x^{2} - 58 x) + 4 (49) = 0$

Factor $4$ out of $4 (9 x^{2} - 58 x) + 4 (49)$ .

$4 (9 x^{2} - 58 x + 49) = 0$

Factor.

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 $a \cdot c = 9 \cdot 49 = 441$ and whose sum is $b = - 58$ .

Factor $- 58$ out of $- 58 x$ .

$4 (9 x^{2} - 58 x + 49) = 0$

Rewrite $- 58$ as $- 9$ plus $- 49$

$4 (9 x^{2} + (- 9 - 49) x + 49) = 0$

Apply the distributive property.

$4 (9 x^{2} - 9 x - 49 x + 49) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((9 x^{2} - 9 x) - 49 x + 49) = 0$

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

$4 (9 x (x - 1) - 49 (x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$4 ((x - 1) (9 x - 49)) = 0$

Remove unnecessary parentheses.

$4 (x - 1) (9 x - 49) = 0$

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

$x - 1 = 0$

$9 x - 49 = 0$

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.

$x = 1$

Set $9 x - 49$ equal to $0$ and solve for $x$ .

Set $9 x - 49$ equal to $0$ .

$9 x - 49 = 0$

Solve $9 x - 49 = 0$ for $x$ .

Add $49$ to both sides of the equation.

$9 x = 49$

Divide each term in $9 x = 49$ by $9$ and simplify.

Divide each term in $9 x = 49$ by $9$ .

$\frac{9 x}{9} = \frac{49}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{49}{9}$

Divide $x$ by $1$ .

$x = \frac{49}{9}$

The final solution is all the values that make $4 (x - 1) (9 x - 49) = 0$ true.

$x = 1, \frac{49}{9}$

Step 8

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

$x = \frac{49}{9}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{49}{9}$

Decimal Form:

$x = 5.\overline{4}$

Mixed Number Form:

$x = 5 \frac{4}{9}$

Solve for x square root of x+4 = square root of 7x+2

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