Solve for x square root of x+7-1 = square root of 3x+10

$\sqrt{x + 7} - 1 = \sqrt{3 x + 10}$

Step 1

Add $1$ to both sides of the equation.

$\sqrt{x + 7} = \sqrt{3 x + 10} + 1$

Step 2

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

${\sqrt{x + 7}}^{2} = {(\sqrt{3 x + 10} + 1)}^{2}$

Step 3

Simplify each side of the equation.

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

${({(x + 7)}^{\frac{1}{2}})}^{2} = {(\sqrt{3 x + 10} + 1)}^{2}$

Simplify the left side.

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

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

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

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(\sqrt{3 x + 10} + 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(\sqrt{3 x + 10} + 1)}^{2}$

Rewrite the expression.

${(x + 7)}^{1} = {(\sqrt{3 x + 10} + 1)}^{2}$

Simplify.

$x + 7 = {(\sqrt{3 x + 10} + 1)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{3 x + 10} + 1)}^{2}$ .

Rewrite ${(\sqrt{3 x + 10} + 1)}^{2}$ as $(\sqrt{3 x + 10} + 1) (\sqrt{3 x + 10} + 1)$ .

$x + 7 = (\sqrt{3 x + 10} + 1) (\sqrt{3 x + 10} + 1)$

Expand $(\sqrt{3 x + 10} + 1) (\sqrt{3 x + 10} + 1)$ using the FOIL Method.

Apply the distributive property.

$x + 7 = \sqrt{3 x + 10} (\sqrt{3 x + 10} + 1) + 1 (\sqrt{3 x + 10} + 1)$

Apply the distributive property.

$x + 7 = \sqrt{3 x + 10} \sqrt{3 x + 10} + \sqrt{3 x + 10} \cdot 1 + 1 (\sqrt{3 x + 10} + 1)$

Apply the distributive property.

$x + 7 = \sqrt{3 x + 10} \sqrt{3 x + 10} + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{3 x + 10} \sqrt{3 x + 10}$ .

Raise $\sqrt{3 x + 10}$ to the power of $1$ .

$x + 7 = {\sqrt{3 x + 10}}^{1} \sqrt{3 x + 10} + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

Raise $\sqrt{3 x + 10}$ to the power of $1$ .

$x + 7 = {\sqrt{3 x + 10}}^{1} {\sqrt{3 x + 10}}^{1} + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

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

$x + 7 = {\sqrt{3 x + 10}}^{1 + 1} + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

Add $1$ and $1$ .

$x + 7 = {\sqrt{3 x + 10}}^{2} + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

Rewrite ${\sqrt{3 x + 10}}^{2}$ as $3 x + 10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 10}$ as ${(3 x + 10)}^{\frac{1}{2}}$ .

$x + 7 = {({(3 x + 10)}^{\frac{1}{2}})}^{2} + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

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

$x + 7 = {(3 x + 10)}^{\frac{1}{2} \cdot 2} + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

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

$x + 7 = {(3 x + 10)}^{\frac{2}{2}} + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 7 = {(3 x + 10)}^{\frac{2}{2}} + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

Rewrite the expression.

$x + 7 = {(3 x + 10)}^{1} + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

Simplify.

$x + 7 = 3 x + 10 + \sqrt{3 x + 10} \cdot 1 + 1 \sqrt{3 x + 10} + 1 \cdot 1$

Multiply $\sqrt{3 x + 10}$ by $1$ .

$x + 7 = 3 x + 10 + \sqrt{3 x + 10} + 1 \sqrt{3 x + 10} + 1 \cdot 1$

Multiply $\sqrt{3 x + 10}$ by $1$ .

$x + 7 = 3 x + 10 + \sqrt{3 x + 10} + \sqrt{3 x + 10} + 1 \cdot 1$

Multiply $1$ by $1$ .

$x + 7 = 3 x + 10 + \sqrt{3 x + 10} + \sqrt{3 x + 10} + 1$

Add $10$ and $1$ .

$x + 7 = 3 x + 11 + \sqrt{3 x + 10} + \sqrt{3 x + 10}$

Add $\sqrt{3 x + 10}$ and $\sqrt{3 x + 10}$ .

$x + 7 = 3 x + 11 + 2 \sqrt{3 x + 10}$

Step 4

Solve for $2 \sqrt{3 x + 10}$ .

Rewrite the equation as $3 x + 11 + 2 \sqrt{3 x + 10} = x + 7$ .

$3 x + 11 + 2 \sqrt{3 x + 10} = x + 7$

Move all terms not containing $2 \sqrt{3 x + 10}$ to the right side of the equation.

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

$11 + 2 \sqrt{3 x + 10} = x + 7 - 3 x$

Subtract $11$ from both sides of the equation.

$2 \sqrt{3 x + 10} = x + 7 - 3 x - 11$

Subtract $3 x$ from $x$ .

$2 \sqrt{3 x + 10} = - 2 x + 7 - 11$

Subtract $11$ from $7$ .

$2 \sqrt{3 x + 10} = - 2 x - 4$

Step 5

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

${(2 \sqrt{3 x + 10})}^{2} = {(- 2 x - 4)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 10}$ as ${(3 x + 10)}^{\frac{1}{2}}$ .

${(2 {(3 x + 10)}^{\frac{1}{2}})}^{2} = {(- 2 x - 4)}^{2}$

Simplify the left side.

Simplify ${(2 {(3 x + 10)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 {(3 x + 10)}^{\frac{1}{2}}$ .

$2^{2} {({(3 x + 10)}^{\frac{1}{2}})}^{2} = {(- 2 x - 4)}^{2}$

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

$4 {({(3 x + 10)}^{\frac{1}{2}})}^{2} = {(- 2 x - 4)}^{2}$

Multiply the exponents in ${({(3 x + 10)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(3 x + 10)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(3 x + 10)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 4)}^{2}$

Rewrite the expression.

$4 {(3 x + 10)}^{1} = {(- 2 x - 4)}^{2}$

Simplify.

$4 (3 x + 10) = {(- 2 x - 4)}^{2}$

Apply the distributive property.

$4 (3 x) + 4 \cdot 10 = {(- 2 x - 4)}^{2}$

Multiply.

Multiply $3$ by $4$ .

$12 x + 4 \cdot 10 = {(- 2 x - 4)}^{2}$

Multiply $4$ by $10$ .

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

Simplify the right side.

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

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

$12 x + 40 = (- 2 x - 4) (- 2 x - 4)$

Expand $(- 2 x - 4) (- 2 x - 4)$ using the FOIL Method.

Apply the distributive property.

$12 x + 40 = - 2 x (- 2 x - 4) - 4 (- 2 x - 4)$

Apply the distributive property.

$12 x + 40 = - 2 x (- 2 x) - 2 x \cdot - 4 - 4 (- 2 x - 4)$

Apply the distributive property.

$12 x + 40 = - 2 x (- 2 x) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$12 x + 40 = - 2 \cdot - 2 x \cdot x - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4$

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

Move $x$ .

$12 x + 40 = - 2 \cdot - 2 (x \cdot x) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4$

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 2$ .

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

Multiply $- 4$ by $- 2$ .

$12 x + 40 = 4 x^{2} + 8 x - 4 (- 2 x) - 4 \cdot - 4$

Multiply $- 2$ by $- 4$ .

$12 x + 40 = 4 x^{2} + 8 x + 8 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$12 x + 40 = 4 x^{2} + 8 x + 8 x + 16$

Add $8 x$ and $8 x$ .

$12 x + 40 = 4 x^{2} + 16 x + 16$

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.

$4 x^{2} + 16 x + 16 = 12 x + 40$

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

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

$4 x^{2} + 16 x + 16 - 12 x = 40$

Subtract $12 x$ from $16 x$ .

$4 x^{2} + 4 x + 16 = 40$

Subtract $40$ from both sides of the equation.

$4 x^{2} + 4 x + 16 - 40 = 0$

Subtract $40$ from $16$ .

$4 x^{2} + 4 x - 24 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} + 4 x - 24$ .

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

$4 (x^{2}) + 4 x - 24 = 0$

Factor $4$ out of $4 x$ .

$4 (x^{2}) + 4 (x) - 24 = 0$

Factor $4$ out of $- 24$ .

$4 (x^{2}) + 4 x + 4 \cdot - 6 = 0$

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

$4 (x^{2} + x) + 4 \cdot - 6 = 0$

Factor $4$ out of $4 (x^{2} + x) + 4 \cdot - 6$ .

$4 (x^{2} + x - 6) = 0$

Factor.

Factor $x^{2} + x - 6$ 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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Write the factored form using these integers.

$4 ((x - 2) (x + 3)) = 0$

Remove unnecessary parentheses.

$4 (x - 2) (x + 3) = 0$

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

$x - 2 = 0$

$x + 3 = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $4 (x - 2) (x + 3) = 0$ true.

$x = 2, - 3$

Step 8

Exclude the solutions that do not make $\sqrt{x + 7} - 1 = \sqrt{3 x + 10}$ true.

$x = - 3$

Solve for x square root of x+7-1 = square root of 3x+10

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