Solve for x 3x=15+3 square root of x+1

$3 x = 15 + 3 \sqrt{x + 1}$

Step 1

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

$15 + 3 \sqrt{x + 1} = 3 x$

Step 2

Subtract $15$ from both sides of the equation.

$3 \sqrt{x + 1} = 3 x - 15$

Step 3

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

${(3 \sqrt{x + 1})}^{2} = {(3 x - 15)}^{2}$

Step 4

Simplify each side of the equation.

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

${(3 {(x + 1)}^{\frac{1}{2}})}^{2} = {(3 x - 15)}^{2}$

Simplify the left side.

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

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

$3^{2} {({(x + 1)}^{\frac{1}{2}})}^{2} = {(3 x - 15)}^{2}$

Raise $3$ to the power of $2$ .

$9 {({(x + 1)}^{\frac{1}{2}})}^{2} = {(3 x - 15)}^{2}$

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

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

$9 {(x + 1)}^{\frac{1}{2} \cdot 2} = {(3 x - 15)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 {(x + 1)}^{\frac{1}{2} \cdot 2} = {(3 x - 15)}^{2}$

Rewrite the expression.

$9 {(x + 1)}^{1} = {(3 x - 15)}^{2}$

Simplify.

$9 (x + 1) = {(3 x - 15)}^{2}$

Apply the distributive property.

$9 x + 9 \cdot 1 = {(3 x - 15)}^{2}$

Multiply $9$ by $1$ .

$9 x + 9 = {(3 x - 15)}^{2}$

Simplify the right side.

Simplify ${(3 x - 15)}^{2}$ .

Rewrite ${(3 x - 15)}^{2}$ as $(3 x - 15) (3 x - 15)$ .

$9 x + 9 = (3 x - 15) (3 x - 15)$

Expand $(3 x - 15) (3 x - 15)$ using the FOIL Method.

Apply the distributive property.

$9 x + 9 = 3 x (3 x - 15) - 15 (3 x - 15)$

Apply the distributive property.

$9 x + 9 = 3 x (3 x) + 3 x \cdot - 15 - 15 (3 x - 15)$

Apply the distributive property.

$9 x + 9 = 3 x (3 x) + 3 x \cdot - 15 - 15 (3 x) - 15 \cdot - 15$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$9 x + 9 = 3 \cdot 3 x \cdot x + 3 x \cdot - 15 - 15 (3 x) - 15 \cdot - 15$

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

Move $x$ .

$9 x + 9 = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 15 - 15 (3 x) - 15 \cdot - 15$

Multiply $x$ by $x$ .

$9 x + 9 = 3 \cdot 3 x^{2} + 3 x \cdot - 15 - 15 (3 x) - 15 \cdot - 15$

Multiply $3$ by $3$ .

$9 x + 9 = 9 x^{2} + 3 x \cdot - 15 - 15 (3 x) - 15 \cdot - 15$

Multiply $- 15$ by $3$ .

$9 x + 9 = 9 x^{2} - 45 x - 15 (3 x) - 15 \cdot - 15$

Multiply $3$ by $- 15$ .

$9 x + 9 = 9 x^{2} - 45 x - 45 x - 15 \cdot - 15$

Multiply $- 15$ by $- 15$ .

$9 x + 9 = 9 x^{2} - 45 x - 45 x + 225$

Subtract $45 x$ from $- 45 x$ .

$9 x + 9 = 9 x^{2} - 90 x + 225$

Step 5

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.

$9 x^{2} - 90 x + 225 = 9 x + 9$

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

Subtract $9 x$ from both sides of the equation.

$9 x^{2} - 90 x + 225 - 9 x = 9$

Subtract $9 x$ from $- 90 x$ .

$9 x^{2} - 99 x + 225 = 9$

Subtract $9$ from both sides of the equation.

$9 x^{2} - 99 x + 225 - 9 = 0$

Subtract $9$ from $225$ .

$9 x^{2} - 99 x + 216 = 0$

Factor the left side of the equation.

Factor $9$ out of $9 x^{2} - 99 x + 216$ .

Factor $9$ out of $9 x^{2}$ .

$9 (x^{2}) - 99 x + 216 = 0$

Factor $9$ out of $- 99 x$ .

$9 (x^{2}) + 9 (- 11 x) + 216 = 0$

Factor $9$ out of $216$ .

$9 x^{2} + 9 (- 11 x) + 9 \cdot 24 = 0$

Factor $9$ out of $9 x^{2} + 9 (- 11 x)$ .

$9 (x^{2} - 11 x) + 9 \cdot 24 = 0$

Factor $9$ out of $9 (x^{2} - 11 x) + 9 \cdot 24$ .

$9 (x^{2} - 11 x + 24) = 0$

Factor.

Factor $x^{2} - 11 x + 24$ 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 $24$ and whose sum is $- 11$ .

$- 8, - 3$

Write the factored form using these integers.

$9 ((x - 8) (x - 3)) = 0$

Remove unnecessary parentheses.

$9 (x - 8) (x - 3) = 0$

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

$x - 8 = 0$

$x - 3 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $9 (x - 8) (x - 3) = 0$ true.

$x = 8, 3$

Step 6

Exclude the solutions that do not make $3 x = 15 + 3 \sqrt{x + 1}$ true.

$x = 8$

Solve for x 3x=15+3 square root of x+1

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