Solve the Rational Equation for x x- square root of 3x-2=4

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

Step 1

Subtract $x$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

Raise $- 1$ to the power of $2$ .

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

Multiply ${({(3 x - 2)}^{\frac{1}{2}})}^{2}$ by $1$ .

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

$3 x - 2 = 4 \cdot 4 + 4 (- x) - x (4 - x)$

Apply the distributive property.

$3 x - 2 = 4 \cdot 4 + 4 (- x) - x \cdot 4 - x (- x)$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$3 x - 2 = 16 + 4 (- x) - x \cdot 4 - x (- x)$

Multiply $- 1$ by $4$ .

$3 x - 2 = 16 - 4 x - x \cdot 4 - x (- x)$

Multiply $4$ by $- 1$ .

$3 x - 2 = 16 - 4 x - 4 x - x (- x)$

Rewrite using the commutative property of multiplication.

$3 x - 2 = 16 - 4 x - 4 x - 1 \cdot - 1 x \cdot x$

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

Move $x$ .

$3 x - 2 = 16 - 4 x - 4 x - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$3 x - 2 = 16 - 4 x - 4 x - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 16 - 4 x - 4 x + 1 x^{2}$

Multiply $x^{2}$ by $1$ .

$3 x - 2 = 16 - 4 x - 4 x + x^{2}$

Subtract $4 x$ from $- 4 x$ .

$3 x - 2 = 16 - 8 x + x^{2}$

Step 4

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.

$16 - 8 x + x^{2} = 3 x - 2$

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

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

$16 - 8 x + x^{2} - 3 x = - 2$

Subtract $3 x$ from $- 8 x$ .

$16 + x^{2} - 11 x = - 2$

Add $2$ to both sides of the equation.

$16 + x^{2} - 11 x + 2 = 0$

Add $16$ and $2$ .

$x^{2} - 11 x + 18 = 0$

Factor $x^{2} - 11 x + 18$ 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 $18$ and whose sum is $- 11$ .

$- 9, - 2$

Write the factored form using these integers.

$(x - 9) (x - 2) = 0$

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

$x - 9 = 0$

$x - 2 = 0$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

$x = 9, 2$

Step 5

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

$x = 9$

Solve the Rational Equation for x x- square root of 3x-2=4

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