Solve for x square root of x-1+ square root of 3x-2-3=0

$\sqrt{x - 1} + \sqrt{3 x - 2} - 3 = 0$

Step 1

Move all terms not containing $\sqrt{x - 1}$ to the right side of the equation.

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

$\sqrt{x - 1} - 3 = - \sqrt{3 x - 2}$

Add $3$ to both sides of the equation.

$\sqrt{x - 1} = - \sqrt{3 x - 2} + 3$

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(- \sqrt{3 x - 2} + 3)}^{2}$ .

Rewrite ${(- \sqrt{3 x - 2} + 3)}^{2}$ as $(- \sqrt{3 x - 2} + 3) (- \sqrt{3 x - 2} + 3)$ .

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

Expand $(- \sqrt{3 x - 2} + 3) (- \sqrt{3 x - 2} + 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- \sqrt{3 x - 2} (- \sqrt{3 x - 2})$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{3 x - 2}$ by $1$ .

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

Raise $\sqrt{3 x - 2}$ to the power of $1$ .

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

Raise $\sqrt{3 x - 2}$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3 x - 2}}^{2}$ as $3 x - 2$ .

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

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Multiply $3$ by $- 1$ .

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

Multiply $- 1$ by $3$ .

$x - 1 = 3 x - 2 - 3 \sqrt{3 x - 2} - 3 \sqrt{3 x - 2} + 3 \cdot 3$

Multiply $3$ by $3$ .

$x - 1 = 3 x - 2 - 3 \sqrt{3 x - 2} - 3 \sqrt{3 x - 2} + 9$

Add $- 2$ and $9$ .

$x - 1 = 3 x + 7 - 3 \sqrt{3 x - 2} - 3 \sqrt{3 x - 2}$

Subtract $3 \sqrt{3 x - 2}$ from $- 3 \sqrt{3 x - 2}$ .

$x - 1 = 3 x + 7 - 6 \sqrt{3 x - 2}$

Step 4

Solve for $- 6 \sqrt{3 x - 2}$ .

Rewrite the equation as $3 x + 7 - 6 \sqrt{3 x - 2} = x - 1$ .

$3 x + 7 - 6 \sqrt{3 x - 2} = x - 1$

Move all terms not containing $- 6 \sqrt{3 x - 2}$ to the right side of the equation.

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

$7 - 6 \sqrt{3 x - 2} = x - 1 - 3 x$

Subtract $7$ from both sides of the equation.

$- 6 \sqrt{3 x - 2} = x - 1 - 3 x - 7$

Subtract $3 x$ from $x$ .

$- 6 \sqrt{3 x - 2} = - 2 x - 1 - 7$

Subtract $7$ from $- 1$ .

$- 6 \sqrt{3 x - 2} = - 2 x - 8$

Step 5

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

${(- 6 \sqrt{3 x - 2})}^{2} = {(- 2 x - 8)}^{2}$

Step 6

Simplify each side of the equation.

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

${(- 6 {(3 x - 2)}^{\frac{1}{2}})}^{2} = {(- 2 x - 8)}^{2}$

Simplify the left side.

Simplify ${(- 6 {(3 x - 2)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 6 {(3 x - 2)}^{\frac{1}{2}}$ .

${(- 6)}^{2} {({(3 x - 2)}^{\frac{1}{2}})}^{2} = {(- 2 x - 8)}^{2}$

Raise $- 6$ to the power of $2$ .

$36 {({(3 x - 2)}^{\frac{1}{2}})}^{2} = {(- 2 x - 8)}^{2}$

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

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

$36 {(3 x - 2)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(3 x - 2)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 8)}^{2}$

Rewrite the expression.

$36 {(3 x - 2)}^{1} = {(- 2 x - 8)}^{2}$

Simplify.

$36 (3 x - 2) = {(- 2 x - 8)}^{2}$

Apply the distributive property.

$36 (3 x) + 36 \cdot - 2 = {(- 2 x - 8)}^{2}$

Multiply.

Multiply $3$ by $36$ .

$108 x + 36 \cdot - 2 = {(- 2 x - 8)}^{2}$

Multiply $36$ by $- 2$ .

$108 x - 72 = {(- 2 x - 8)}^{2}$

Simplify the right side.

Simplify ${(- 2 x - 8)}^{2}$ .

Rewrite ${(- 2 x - 8)}^{2}$ as $(- 2 x - 8) (- 2 x - 8)$ .

$108 x - 72 = (- 2 x - 8) (- 2 x - 8)$

Expand $(- 2 x - 8) (- 2 x - 8)$ using the FOIL Method.

Apply the distributive property.

$108 x - 72 = - 2 x (- 2 x - 8) - 8 (- 2 x - 8)$

Apply the distributive property.

$108 x - 72 = - 2 x (- 2 x) - 2 x \cdot - 8 - 8 (- 2 x - 8)$

Apply the distributive property.

$108 x - 72 = - 2 x (- 2 x) - 2 x \cdot - 8 - 8 (- 2 x) - 8 \cdot - 8$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$108 x - 72 = - 2 \cdot - 2 x \cdot x - 2 x \cdot - 8 - 8 (- 2 x) - 8 \cdot - 8$

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

Move $x$ .

$108 x - 72 = - 2 \cdot - 2 (x \cdot x) - 2 x \cdot - 8 - 8 (- 2 x) - 8 \cdot - 8$

Multiply $x$ by $x$ .

$108 x - 72 = - 2 \cdot - 2 x^{2} - 2 x \cdot - 8 - 8 (- 2 x) - 8 \cdot - 8$

Multiply $- 2$ by $- 2$ .

$108 x - 72 = 4 x^{2} - 2 x \cdot - 8 - 8 (- 2 x) - 8 \cdot - 8$

Multiply $- 8$ by $- 2$ .

$108 x - 72 = 4 x^{2} + 16 x - 8 (- 2 x) - 8 \cdot - 8$

Multiply $- 2$ by $- 8$ .

$108 x - 72 = 4 x^{2} + 16 x + 16 x - 8 \cdot - 8$

Multiply $- 8$ by $- 8$ .

$108 x - 72 = 4 x^{2} + 16 x + 16 x + 64$

Add $16 x$ and $16 x$ .

$108 x - 72 = 4 x^{2} + 32 x + 64$

Step 7

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.

$4 x^{2} + 32 x + 64 = 108 x - 72$

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

Subtract $108 x$ from both sides of the equation.

$4 x^{2} + 32 x + 64 - 108 x = - 72$

Subtract $108 x$ from $32 x$ .

$4 x^{2} - 76 x + 64 = - 72$

Add $72$ to both sides of the equation.

$4 x^{2} - 76 x + 64 + 72 = 0$

Add $64$ and $72$ .

$4 x^{2} - 76 x + 136 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 76 x + 136$ .

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) - 76 x + 136 = 0$

Factor $4$ out of $- 76 x$ .

$4 (x^{2}) + 4 (- 19 x) + 136 = 0$

Factor $4$ out of $136$ .

$4 x^{2} + 4 (- 19 x) + 4 \cdot 34 = 0$

Factor $4$ out of $4 x^{2} + 4 (- 19 x)$ .

$4 (x^{2} - 19 x) + 4 \cdot 34 = 0$

Factor $4$ out of $4 (x^{2} - 19 x) + 4 \cdot 34$ .

$4 (x^{2} - 19 x + 34) = 0$

Factor.

Factor $x^{2} - 19 x + 34$ 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 $34$ and whose sum is $- 19$ .

$- 17, - 2$

Write the factored form using these integers.

$4 ((x - 17) (x - 2)) = 0$

Remove unnecessary parentheses.

$4 (x - 17) (x - 2) = 0$

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

$x - 17 = 0$

$x - 2 = 0$

Set $x - 17$ equal to $0$ and solve for $x$ .

Set $x - 17$ equal to $0$ .

$x - 17 = 0$

Add $17$ to both sides of the equation.

$x = 17$

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 $4 (x - 17) (x - 2) = 0$ true.

$x = 17, 2$

Step 8

Exclude the solutions that do not make $\sqrt{x - 1} + \sqrt{3 x - 2} - 3 = 0$ true.

$x = 2$

Solve for x square root of x-1+ square root of 3x-2-3=0

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