Solve for x 4 square root of 3x-2=3x-7

$4 \sqrt{3 x - 2} = 3 x - 7$

Step 1

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

${(4 \sqrt{3 x - 2})}^{2} = {(3 x - 7)}^{2}$

Step 2

Simplify each side of the equation.

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

${(4 {(3 x - 2)}^{\frac{1}{2}})}^{2} = {(3 x - 7)}^{2}$

Simplify the left side.

Simplify ${(4 {(3 x - 2)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $4 {(3 x - 2)}^{\frac{1}{2}}$ .

$4^{2} {({(3 x - 2)}^{\frac{1}{2}})}^{2} = {(3 x - 7)}^{2}$

Raise $4$ to the power of $2$ .

$16 {({(3 x - 2)}^{\frac{1}{2}})}^{2} = {(3 x - 7)}^{2}$

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

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

$16 {(3 x - 2)}^{\frac{1}{2} \cdot 2} = {(3 x - 7)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(3 x - 2)}^{\frac{1}{2} \cdot 2} = {(3 x - 7)}^{2}$

Rewrite the expression.

$16 {(3 x - 2)}^{1} = {(3 x - 7)}^{2}$

Simplify.

$16 (3 x - 2) = {(3 x - 7)}^{2}$

Apply the distributive property.

$16 (3 x) + 16 \cdot - 2 = {(3 x - 7)}^{2}$

Multiply.

Multiply $3$ by $16$ .

$48 x + 16 \cdot - 2 = {(3 x - 7)}^{2}$

Multiply $16$ by $- 2$ .

$48 x - 32 = {(3 x - 7)}^{2}$

Simplify the right side.

Simplify ${(3 x - 7)}^{2}$ .

Rewrite ${(3 x - 7)}^{2}$ as $(3 x - 7) (3 x - 7)$ .

$48 x - 32 = (3 x - 7) (3 x - 7)$

Expand $(3 x - 7) (3 x - 7)$ using the FOIL Method.

Apply the distributive property.

$48 x - 32 = 3 x (3 x - 7) - 7 (3 x - 7)$

Apply the distributive property.

$48 x - 32 = 3 x (3 x) + 3 x \cdot - 7 - 7 (3 x - 7)$

Apply the distributive property.

$48 x - 32 = 3 x (3 x) + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$48 x - 32 = 3 \cdot 3 x \cdot x + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7$

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

Move $x$ .

$48 x - 32 = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7$

Multiply $x$ by $x$ .

$48 x - 32 = 3 \cdot 3 x^{2} + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7$

Multiply $3$ by $3$ .

$48 x - 32 = 9 x^{2} + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7$

Multiply $- 7$ by $3$ .

$48 x - 32 = 9 x^{2} - 21 x - 7 (3 x) - 7 \cdot - 7$

Multiply $3$ by $- 7$ .

$48 x - 32 = 9 x^{2} - 21 x - 21 x - 7 \cdot - 7$

Multiply $- 7$ by $- 7$ .

$48 x - 32 = 9 x^{2} - 21 x - 21 x + 49$

Subtract $21 x$ from $- 21 x$ .

$48 x - 32 = 9 x^{2} - 42 x + 49$

Step 3

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.

$9 x^{2} - 42 x + 49 = 48 x - 32$

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

Subtract $48 x$ from both sides of the equation.

$9 x^{2} - 42 x + 49 - 48 x = - 32$

Subtract $48 x$ from $- 42 x$ .

$9 x^{2} - 90 x + 49 = - 32$

Add $32$ to both sides of the equation.

$9 x^{2} - 90 x + 49 + 32 = 0$

Add $49$ and $32$ .

$9 x^{2} - 90 x + 81 = 0$

Factor the left side of the equation.

Factor $9$ out of $9 x^{2} - 90 x + 81$ .

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

$9 (x^{2}) - 90 x + 81 = 0$

Factor $9$ out of $- 90 x$ .

$9 (x^{2}) + 9 (- 10 x) + 81 = 0$

Factor $9$ out of $81$ .

$9 x^{2} + 9 (- 10 x) + 9 \cdot 9 = 0$

Factor $9$ out of $9 x^{2} + 9 (- 10 x)$ .

$9 (x^{2} - 10 x) + 9 \cdot 9 = 0$

Factor $9$ out of $9 (x^{2} - 10 x) + 9 \cdot 9$ .

$9 (x^{2} - 10 x + 9) = 0$

Factor.

Factor $x^{2} - 10 x + 9$ 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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

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

$x - 9 = 0$

$x - 1 = 0$

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

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

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

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.

$x = 1$

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

$x = 9, 1$

Step 4

Exclude the solutions that do not make $4 \sqrt{3 x - 2} = 3 x - 7$ true.

$x = 9$

Solve for x 4 square root of 3x-2=3x-7

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