Solve for z square root of z+4+1 = square root of 3z+1

$\sqrt{z + 4} + 1 = \sqrt{3 z + 1}$

Subtract $1$ from both sides of the equation.

$\sqrt{z + 4} = \sqrt{3 z + 1} - 1$

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

${\sqrt{z + 4}}^{2} = {(\sqrt{3 z + 1} - 1)}^{2}$

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(z + 4)}^{1} = {(\sqrt{3 z + 1} - 1)}^{2}$

Simplify.

$z + 4 = {(\sqrt{3 z + 1} - 1)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{3 z + 1} - 1)}^{2}$ .

Rewrite ${(\sqrt{3 z + 1} - 1)}^{2}$ as $(\sqrt{3 z + 1} - 1) (\sqrt{3 z + 1} - 1)$ .

$z + 4 = (\sqrt{3 z + 1} - 1) (\sqrt{3 z + 1} - 1)$

Expand $(\sqrt{3 z + 1} - 1) (\sqrt{3 z + 1} - 1)$ using the FOIL Method.

Apply the distributive property.

$z + 4 = \sqrt{3 z + 1} (\sqrt{3 z + 1} - 1) - 1 (\sqrt{3 z + 1} - 1)$

Apply the distributive property.

$z + 4 = \sqrt{3 z + 1} \sqrt{3 z + 1} + \sqrt{3 z + 1} \cdot - 1 - 1 (\sqrt{3 z + 1} - 1)$

Apply the distributive property.

$z + 4 = \sqrt{3 z + 1} \sqrt{3 z + 1} + \sqrt{3 z + 1} \cdot - 1 - 1 \sqrt{3 z + 1} - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{3 z + 1} \sqrt{3 z + 1}$ .

Raise $\sqrt{3 z + 1}$ to the power of $1$ .

$z + 4 = {\sqrt{3 z + 1}}^{1} \sqrt{3 z + 1} + \sqrt{3 z + 1} \cdot - 1 - 1 \sqrt{3 z + 1} - 1 \cdot - 1$

Raise $\sqrt{3 z + 1}$ to the power of $1$ .

$z + 4 = {\sqrt{3 z + 1}}^{1} {\sqrt{3 z + 1}}^{1} + \sqrt{3 z + 1} \cdot - 1 - 1 \sqrt{3 z + 1} - 1 \cdot - 1$

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

$z + 4 = {\sqrt{3 z + 1}}^{1 + 1} + \sqrt{3 z + 1} \cdot - 1 - 1 \sqrt{3 z + 1} - 1 \cdot - 1$

Add $1$ and $1$ .

$z + 4 = {\sqrt{3 z + 1}}^{2} + \sqrt{3 z + 1} \cdot - 1 - 1 \sqrt{3 z + 1} - 1 \cdot - 1$

Rewrite ${\sqrt{3 z + 1}}^{2}$ as $3 z + 1$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$z + 4 = {(3 z + 1)}^{1} + \sqrt{3 z + 1} \cdot - 1 - 1 \sqrt{3 z + 1} - 1 \cdot - 1$

Simplify.

$z + 4 = 3 z + 1 + \sqrt{3 z + 1} \cdot - 1 - 1 \sqrt{3 z + 1} - 1 \cdot - 1$

Move $- 1$ to the left of $\sqrt{3 z + 1}$ .

$z + 4 = 3 z + 1 - 1 \cdot \sqrt{3 z + 1} - 1 \sqrt{3 z + 1} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{3 z + 1}$ as $- \sqrt{3 z + 1}$ .

$z + 4 = 3 z + 1 - \sqrt{3 z + 1} - 1 \sqrt{3 z + 1} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{3 z + 1}$ as $- \sqrt{3 z + 1}$ .

$z + 4 = 3 z + 1 - \sqrt{3 z + 1} - \sqrt{3 z + 1} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$z + 4 = 3 z + 1 - \sqrt{3 z + 1} - \sqrt{3 z + 1} + 1$

Add $1$ and $1$ .

$z + 4 = 3 z + 2 - \sqrt{3 z + 1} - \sqrt{3 z + 1}$

Subtract $\sqrt{3 z + 1}$ from $- \sqrt{3 z + 1}$ .

$z + 4 = 3 z + 2 - 2 \sqrt{3 z + 1}$

Solve for $- 2 \sqrt{3 z + 1}$ .

Rewrite the equation as $3 z + 2 - 2 \sqrt{3 z + 1} = z + 4$ .

$3 z + 2 - 2 \sqrt{3 z + 1} = z + 4$

Move all terms not containing $- 2 \sqrt{3 z + 1}$ to the right side of the equation.

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

$2 - 2 \sqrt{3 z + 1} = z + 4 - 3 z$

Subtract $2$ from both sides of the equation.

$- 2 \sqrt{3 z + 1} = z + 4 - 3 z - 2$

Subtract $3 z$ from $z$ .

$- 2 \sqrt{3 z + 1} = - 2 z + 4 - 2$

Subtract $2$ from $4$ .

$- 2 \sqrt{3 z + 1} = - 2 z + 2$

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

${(- 2 \sqrt{3 z + 1})}^{2} = {(- 2 z + 2)}^{2}$

Simplify each side of the equation.

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

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

Simplify the left side.

Simplify ${(- 2 {(3 z + 1)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 2 {(3 z + 1)}^{\frac{1}{2}}$ .

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

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

$4 {({(3 z + 1)}^{\frac{1}{2}})}^{2} = {(- 2 z + 2)}^{2}$

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$4 {(3 z + 1)}^{1} = {(- 2 z + 2)}^{2}$

Simplify.

$4 (3 z + 1) = {(- 2 z + 2)}^{2}$

Apply the distributive property.

$4 (3 z) + 4 \cdot 1 = {(- 2 z + 2)}^{2}$

Multiply.

Multiply $3$ by $4$ .

$12 z + 4 \cdot 1 = {(- 2 z + 2)}^{2}$

Multiply $4$ by $1$ .

$12 z + 4 = {(- 2 z + 2)}^{2}$

Simplify the right side.

Simplify ${(- 2 z + 2)}^{2}$ .

Rewrite ${(- 2 z + 2)}^{2}$ as $(- 2 z + 2) (- 2 z + 2)$ .

$12 z + 4 = (- 2 z + 2) (- 2 z + 2)$

Expand $(- 2 z + 2) (- 2 z + 2)$ using the FOIL Method.

Apply the distributive property.

$12 z + 4 = - 2 z (- 2 z + 2) + 2 (- 2 z + 2)$

Apply the distributive property.

$12 z + 4 = - 2 z (- 2 z) - 2 z \cdot 2 + 2 (- 2 z + 2)$

Apply the distributive property.

$12 z + 4 = - 2 z (- 2 z) - 2 z \cdot 2 + 2 (- 2 z) + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$12 z + 4 = - 2 \cdot - 2 z \cdot z - 2 z \cdot 2 + 2 (- 2 z) + 2 \cdot 2$

Multiply $z$ by $z$ by adding the exponents.

Move $z$ .

$12 z + 4 = - 2 \cdot - 2 (z \cdot z) - 2 z \cdot 2 + 2 (- 2 z) + 2 \cdot 2$

Multiply $z$ by $z$ .

$12 z + 4 = - 2 \cdot - 2 z^{2} - 2 z \cdot 2 + 2 (- 2 z) + 2 \cdot 2$

Multiply $- 2$ by $- 2$ .

$12 z + 4 = 4 z^{2} - 2 z \cdot 2 + 2 (- 2 z) + 2 \cdot 2$

Multiply $2$ by $- 2$ .

$12 z + 4 = 4 z^{2} - 4 z + 2 (- 2 z) + 2 \cdot 2$

Multiply $- 2$ by $2$ .

$12 z + 4 = 4 z^{2} - 4 z - 4 z + 2 \cdot 2$

Multiply $2$ by $2$ .

$12 z + 4 = 4 z^{2} - 4 z - 4 z + 4$

Subtract $4 z$ from $- 4 z$ .

$12 z + 4 = 4 z^{2} - 8 z + 4$

Solve for $z$ .

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

$4 z^{2} - 8 z + 4 = 12 z + 4$

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

Subtract $12 z$ from both sides of the equation.

$4 z^{2} - 8 z + 4 - 12 z = 4$

Subtract $12 z$ from $- 8 z$ .

$4 z^{2} - 20 z + 4 = 4$

Subtract $4$ from both sides of the equation.

$4 z^{2} - 20 z + 4 - 4 = 0$

Combine the opposite terms in $4 z^{2} - 20 z + 4 - 4$ .

Subtract $4$ from $4$ .

$4 z^{2} - 20 z + 0 = 0$

Add $4 z^{2} - 20 z$ and $0$ .

$4 z^{2} - 20 z = 0$

Factor $4 z$ out of $4 z^{2} - 20 z$ .

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

$4 z (z) - 20 z = 0$

Factor $4 z$ out of $- 20 z$ .

$4 z (z) + 4 z (- 5) = 0$

Factor $4 z$ out of $4 z (z) + 4 z (- 5)$ .

$4 z (z - 5) = 0$

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

$z = 0$

$z - 5 = 0$

Set $z$ equal to $0$ .

$z = 0$

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

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

$z - 5 = 0$

Add $5$ to both sides of the equation.

$z = 5$

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

$z = 0, 5$

Exclude the solutions that do not make $\sqrt{z + 4} + 1 = \sqrt{3 z + 1}$ true.

$z = 5$

Solve for z square root of z+4+1 = square root of 3z+1

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