Solve for x x-2 = square root of 3/2x-2

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

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.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\frac{3}{2}$ and $x$ .

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

$\frac{3 x}{2} - 2 = x \cdot x + x \cdot - 2 - 2 (x - 2)$

Apply the distributive property.

$\frac{3 x}{2} - 2 = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{3 x}{2} - 2 = x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2$

Move $- 2$ to the left of $x$ .

$\frac{3 x}{2} - 2 = x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$\frac{3 x}{2} - 2 = x^{2} - 2 x - 2 x + 4$

Subtract $2 x$ from $- 2 x$ .

$\frac{3 x}{2} - 2 = x^{2} - 4 x + 4$

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.

$x^{2} - 4 x + 4 = \frac{3 x}{2} - 2$

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

Subtract $\frac{3 x}{2}$ from both sides of the equation.

$x^{2} - 4 x + 4 - \frac{3 x}{2} = - 2$

To write $- 4 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} - 4 x \cdot \frac{2}{2} - \frac{3 x}{2} + 4 = - 2$

Combine $- 4 x$ and $\frac{2}{2}$ .

$x^{2} + \frac{- 4 x \cdot 2}{2} - \frac{3 x}{2} + 4 = - 2$

Combine the numerators over the common denominator.

$x^{2} + \frac{- 4 x \cdot 2 - 3 x}{2} + 4 = - 2$

Find the common denominator.

Write $x^{2}$ as a fraction with denominator $1$ .

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

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

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

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

$\frac{x^{2} \cdot 2}{2} + \frac{- 4 x \cdot 2 - 3 x}{2} + 4 = - 2$

Write $4$ as a fraction with denominator $1$ .

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

Multiply $\frac{4}{1}$ by $\frac{2}{2}$ .

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

Multiply $\frac{4}{1}$ by $\frac{2}{2}$ .

$\frac{x^{2} \cdot 2}{2} + \frac{- 4 x \cdot 2 - 3 x}{2} + \frac{4 \cdot 2}{2} = - 2$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 2 - 4 x \cdot 2 - 3 x + 4 \cdot 2}{2} = - 2$

Simplify each term.

Move $2$ to the left of $x^{2}$ .

$\frac{2 \cdot x^{2} - 4 x \cdot 2 - 3 x + 4 \cdot 2}{2} = - 2$

Multiply $2$ by $- 4$ .

$\frac{2 x^{2} - 8 x - 3 x + 4 \cdot 2}{2} = - 2$

Multiply $4$ by $2$ .

$\frac{2 x^{2} - 8 x - 3 x + 8}{2} = - 2$

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

$\frac{2 x^{2} - 11 x + 8}{2} = - 2$

Multiply both sides by $2$ .

$\frac{2 x^{2} - 11 x + 8}{2} \cdot 2 = - 2 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2} - 11 x + 8}{2} \cdot 2 = - 2 \cdot 2$

Rewrite the expression.

$2 x^{2} - 11 x + 8 = - 2 \cdot 2$

Simplify the right side.

Multiply $- 2$ by $2$ .

$2 x^{2} - 11 x + 8 = - 4$

Solve for $x$ .

Add $4$ to both sides of the equation.

$2 x^{2} - 11 x + 8 + 4 = 0$

Add $8$ and $4$ .

$2 x^{2} - 11 x + 12 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 12 = 24$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 x$ .

$2 x^{2} - 11 x + 12 = 0$

Rewrite $- 11$ as $- 3$ plus $- 8$

$2 x^{2} + (- 3 - 8) x + 12 = 0$

Apply the distributive property.

$2 x^{2} - 3 x - 8 x + 12 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{2} - 3 x) - 8 x + 12 = 0$

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

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

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

$2 x - 3 = 0$

$x - 4 = 0$

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Solve $2 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

$x = \frac{3}{2}, 4$

Step 5

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

$x = 4$

Solve for x x-2 = square root of 3/2x-2

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