Solve by Factoring square root of 3x+7=3x+5

$\sqrt{3 x + 7} = 3 x + 5$

Step 1

Move all the expressions to the left side of the equation.

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

$\sqrt{3 x + 7} - 3 x = 5$

Subtract $5$ from both sides of the equation.

$\sqrt{3 x + 7} - 3 x - 5 = 0$

Step 2

Move all terms not containing $\sqrt{3 x + 7}$ to the right side of the equation.

Add $3 x$ to both sides of the equation.

$\sqrt{3 x + 7} - 5 = 3 x$

Add $5$ to both sides of the equation.

$\sqrt{3 x + 7} = 3 x + 5$

Step 3

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

${\sqrt{3 x + 7}}^{2} = {(3 x + 5)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 7}$ as ${(3 x + 7)}^{\frac{1}{2}}$ .

${({(3 x + 7)}^{\frac{1}{2}})}^{2} = {(3 x + 5)}^{2}$

Simplify the left side.

Simplify ${({(3 x + 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x + 7)}^{\frac{1}{2}})}^{2}$ .

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

${(3 x + 7)}^{\frac{1}{2} \cdot 2} = {(3 x + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x + 7)}^{\frac{1}{2} \cdot 2} = {(3 x + 5)}^{2}$

Rewrite the expression.

${(3 x + 7)}^{1} = {(3 x + 5)}^{2}$

Simplify.

$3 x + 7 = {(3 x + 5)}^{2}$

Simplify the right side.

Simplify ${(3 x + 5)}^{2}$ .

Rewrite ${(3 x + 5)}^{2}$ as $(3 x + 5) (3 x + 5)$ .

$3 x + 7 = (3 x + 5) (3 x + 5)$

Expand $(3 x + 5) (3 x + 5)$ using the FOIL Method.

Apply the distributive property.

$3 x + 7 = 3 x (3 x + 5) + 5 (3 x + 5)$

Apply the distributive property.

$3 x + 7 = 3 x (3 x) + 3 x \cdot 5 + 5 (3 x + 5)$

Apply the distributive property.

$3 x + 7 = 3 x (3 x) + 3 x \cdot 5 + 5 (3 x) + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 x + 7 = 3 \cdot 3 x \cdot x + 3 x \cdot 5 + 5 (3 x) + 5 \cdot 5$

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

Move $x$ .

$3 x + 7 = 3 \cdot 3 (x \cdot x) + 3 x \cdot 5 + 5 (3 x) + 5 \cdot 5$

Multiply $x$ by $x$ .

$3 x + 7 = 3 \cdot 3 x^{2} + 3 x \cdot 5 + 5 (3 x) + 5 \cdot 5$

Multiply $3$ by $3$ .

$3 x + 7 = 9 x^{2} + 3 x \cdot 5 + 5 (3 x) + 5 \cdot 5$

Multiply $5$ by $3$ .

$3 x + 7 = 9 x^{2} + 15 x + 5 (3 x) + 5 \cdot 5$

Multiply $3$ by $5$ .

$3 x + 7 = 9 x^{2} + 15 x + 15 x + 5 \cdot 5$

Multiply $5$ by $5$ .

$3 x + 7 = 9 x^{2} + 15 x + 15 x + 25$

Add $15 x$ and $15 x$ .

$3 x + 7 = 9 x^{2} + 30 x + 25$

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.

$9 x^{2} + 30 x + 25 = 3 x + 7$

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

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

$9 x^{2} + 30 x + 25 - 3 x = 7$

Subtract $3 x$ from $30 x$ .

$9 x^{2} + 27 x + 25 = 7$

Subtract $7$ from both sides of the equation.

$9 x^{2} + 27 x + 25 - 7 = 0$

Subtract $7$ from $25$ .

$9 x^{2} + 27 x + 18 = 0$

Factor the left side of the equation.

Factor $9$ out of $9 x^{2} + 27 x + 18$ .

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

$9 (x^{2}) + 27 x + 18 = 0$

Factor $9$ out of $27 x$ .

$9 (x^{2}) + 9 (3 x) + 18 = 0$

Factor $9$ out of $18$ .

$9 x^{2} + 9 (3 x) + 9 \cdot 2 = 0$

Factor $9$ out of $9 x^{2} + 9 (3 x)$ .

$9 (x^{2} + 3 x) + 9 \cdot 2 = 0$

Factor $9$ out of $9 (x^{2} + 3 x) + 9 \cdot 2$ .

$9 (x^{2} + 3 x + 2) = 0$

Factor.

Factor $x^{2} + 3 x + 2$ 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 $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

$9 ((x + 1) (x + 2)) = 0$

Remove unnecessary parentheses.

$9 (x + 1) (x + 2) = 0$

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

$x + 1 = 0$

$x + 2 = 0$

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.

$x = - 1$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $9 (x + 1) (x + 2) = 0$ true.

$x = - 1, - 2$

Step 6

Exclude the solutions that do not make $\sqrt{3 x + 7} - 3 x - 5 = 0$ true.

$x = - 1$

Solve by Factoring square root of 3x+7=3x+5

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