Solve for x square root of x+1 = square root of 6x-6

$\sqrt{x} + 1 = \sqrt{6 x - 6}$

Solve for $\sqrt{x}$ .

Simplify $\sqrt{6 x - 6}$ .

Rewrite.

$\sqrt{x} + 1 = 0 + 0 + \sqrt{6 x - 6}$

Simplify by adding zeros.

$\sqrt{x} + 1 = \sqrt{6 x - 6}$

Factor $6$ out of $6 x - 6$ .

Factor $6$ out of $6 x$ .

$\sqrt{x} + 1 = \sqrt{6 (x) - 6}$

Factor $6$ out of $- 6$ .

$\sqrt{x} + 1 = \sqrt{6 (x) + 6 (- 1)}$

Factor $6$ out of $6 (x) + 6 (- 1)$ .

$\sqrt{x} + 1 = \sqrt{6 (x - 1)}$

Subtract $1$ from both sides of the equation.

$\sqrt{x} = \sqrt{6 (x - 1)} - 1$

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

${\sqrt{x}}^{2} = {(\sqrt{6 (x - 1)} - 1)}^{2}$

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{6 (x - 1)} - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{6 (x - 1)} - 1)}^{2}$

Rewrite the expression.

$x^{1} = {(\sqrt{6 (x - 1)} - 1)}^{2}$

Simplify.

$x = {(\sqrt{6 (x - 1)} - 1)}^{2}$

Simplify the right side.

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

Rewrite ${(\sqrt{6 (x - 1)} - 1)}^{2}$ as $(\sqrt{6 (x - 1)} - 1) (\sqrt{6 (x - 1)} - 1)$ .

$x = (\sqrt{6 (x - 1)} - 1) (\sqrt{6 (x - 1)} - 1)$

Expand $(\sqrt{6 (x - 1)} - 1) (\sqrt{6 (x - 1)} - 1)$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{6 (x - 1)} (\sqrt{6 (x - 1)} - 1) - 1 (\sqrt{6 (x - 1)} - 1)$

Apply the distributive property.

$x = \sqrt{6 (x - 1)} \sqrt{6 (x - 1)} + \sqrt{6 (x - 1)} \cdot - 1 - 1 (\sqrt{6 (x - 1)} - 1)$

Apply the distributive property.

$x = \sqrt{6 (x - 1)} \sqrt{6 (x - 1)} + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{6 (x - 1)} \sqrt{6 (x - 1)}$ .

Raise $\sqrt{6 (x - 1)}$ to the power of $1$ .

$x = {\sqrt{6 (x - 1)}}^{1} \sqrt{6 (x - 1)} + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Raise $\sqrt{6 (x - 1)}$ to the power of $1$ .

$x = {\sqrt{6 (x - 1)}}^{1} {\sqrt{6 (x - 1)}}^{1} + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

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

$x = {\sqrt{6 (x - 1)}}^{1 + 1} + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Add $1$ and $1$ .

$x = {\sqrt{6 (x - 1)}}^{2} + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Rewrite ${\sqrt{6 (x - 1)}}^{2}$ as $6 (x - 1)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 (x - 1)}$ as ${(6 (x - 1))}^{\frac{1}{2}}$ .

$x = {({(6 (x - 1))}^{\frac{1}{2}})}^{2} + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

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

$x = {(6 (x - 1))}^{\frac{1}{2} \cdot 2} + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

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

$x = {(6 (x - 1))}^{\frac{2}{2}} + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = {(6 (x - 1))}^{\frac{2}{2}} + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Rewrite the expression.

$x = {(6 (x - 1))}^{1} + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Simplify.

$x = 6 (x - 1) + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Apply the distributive property.

$x = 6 x + 6 \cdot - 1 + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Multiply $6$ by $- 1$ .

$x = 6 x - 6 + \sqrt{6 (x - 1)} \cdot - 1 - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Move $- 1$ to the left of $\sqrt{6 (x - 1)}$ .

$x = 6 x - 6 - 1 \cdot \sqrt{6 (x - 1)} - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{6 (x - 1)}$ as $- \sqrt{6 (x - 1)}$ .

$x = 6 x - 6 - \sqrt{6 (x - 1)} - 1 \sqrt{6 (x - 1)} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{6 (x - 1)}$ as $- \sqrt{6 (x - 1)}$ .

$x = 6 x - 6 - \sqrt{6 (x - 1)} - \sqrt{6 (x - 1)} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$x = 6 x - 6 - \sqrt{6 (x - 1)} - \sqrt{6 (x - 1)} + 1$

Add $- 6$ and $1$ .

$x = 6 x - 5 - \sqrt{6 (x - 1)} - \sqrt{6 (x - 1)}$

Subtract $\sqrt{6 (x - 1)}$ from $- \sqrt{6 (x - 1)}$ .

$x = 6 x - 5 - 2 \sqrt{6 (x - 1)}$

Solve for $- 2 \sqrt{6 (x - 1)}$ .

Rewrite the equation as $6 x - 5 - 2 \sqrt{6 (x - 1)} = x$ .

$6 x - 5 - 2 \sqrt{6 (x - 1)} = x$

Move all terms not containing $- 2 \sqrt{6 (x - 1)}$ to the right side of the equation.

Subtract $6 x$ from both sides of the equation.

$- 5 - 2 \sqrt{6 (x - 1)} = x - 6 x$

Add $5$ to both sides of the equation.

$- 2 \sqrt{6 (x - 1)} = x - 6 x + 5$

Subtract $6 x$ from $x$ .

$- 2 \sqrt{6 (x - 1)} = - 5 x + 5$

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

${(- 2 \sqrt{6 (x - 1)})}^{2} = {(- 5 x + 5)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 (x - 1)}$ as ${(6 (x - 1))}^{\frac{1}{2}}$ .

${(- 2 {(6 (x - 1))}^{\frac{1}{2}})}^{2} = {(- 5 x + 5)}^{2}$

Simplify the left side.

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

Apply the distributive property.

${(- 2 {(6 x + 6 \cdot - 1)}^{\frac{1}{2}})}^{2} = {(- 5 x + 5)}^{2}$

Simplify the expression.

Multiply $6$ by $- 1$ .

${(- 2 {(6 x - 6)}^{\frac{1}{2}})}^{2} = {(- 5 x + 5)}^{2}$

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

${(- 2)}^{2} {({(6 x - 6)}^{\frac{1}{2}})}^{2} = {(- 5 x + 5)}^{2}$

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

$4 {({(6 x - 6)}^{\frac{1}{2}})}^{2} = {(- 5 x + 5)}^{2}$

Multiply the exponents in ${({(6 x - 6)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(6 x - 6)}^{\frac{1}{2} \cdot 2} = {(- 5 x + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(6 x - 6)}^{\frac{1}{2} \cdot 2} = {(- 5 x + 5)}^{2}$

Rewrite the expression.

$4 {(6 x - 6)}^{1} = {(- 5 x + 5)}^{2}$

Simplify.

$4 (6 x - 6) = {(- 5 x + 5)}^{2}$

Apply the distributive property.

$4 (6 x) + 4 \cdot - 6 = {(- 5 x + 5)}^{2}$

Multiply.

Multiply $6$ by $4$ .

$24 x + 4 \cdot - 6 = {(- 5 x + 5)}^{2}$

Multiply $4$ by $- 6$ .

$24 x - 24 = {(- 5 x + 5)}^{2}$

Simplify the right side.

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

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

$24 x - 24 = (- 5 x + 5) (- 5 x + 5)$

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

Apply the distributive property.

$24 x - 24 = - 5 x (- 5 x + 5) + 5 (- 5 x + 5)$

Apply the distributive property.

$24 x - 24 = - 5 x (- 5 x) - 5 x \cdot 5 + 5 (- 5 x + 5)$

Apply the distributive property.

$24 x - 24 = - 5 x (- 5 x) - 5 x \cdot 5 + 5 (- 5 x) + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$24 x - 24 = - 5 \cdot - 5 x \cdot x - 5 x \cdot 5 + 5 (- 5 x) + 5 \cdot 5$

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

Move $x$ .

$24 x - 24 = - 5 \cdot - 5 (x \cdot x) - 5 x \cdot 5 + 5 (- 5 x) + 5 \cdot 5$

Multiply $x$ by $x$ .

$24 x - 24 = - 5 \cdot - 5 x^{2} - 5 x \cdot 5 + 5 (- 5 x) + 5 \cdot 5$

Multiply $- 5$ by $- 5$ .

$24 x - 24 = 25 x^{2} - 5 x \cdot 5 + 5 (- 5 x) + 5 \cdot 5$

Multiply $5$ by $- 5$ .

$24 x - 24 = 25 x^{2} - 25 x + 5 (- 5 x) + 5 \cdot 5$

Multiply $- 5$ by $5$ .

$24 x - 24 = 25 x^{2} - 25 x - 25 x + 5 \cdot 5$

Multiply $5$ by $5$ .

$24 x - 24 = 25 x^{2} - 25 x - 25 x + 25$

Subtract $25 x$ from $- 25 x$ .

$24 x - 24 = 25 x^{2} - 50 x + 25$

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.

$25 x^{2} - 50 x + 25 = 24 x - 24$

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

Subtract $24 x$ from both sides of the equation.

$25 x^{2} - 50 x + 25 - 24 x = - 24$

Subtract $24 x$ from $- 50 x$ .

$25 x^{2} - 74 x + 25 = - 24$

Add $24$ to both sides of the equation.

$25 x^{2} - 74 x + 25 + 24 = 0$

Add $25$ and $24$ .

$25 x^{2} - 74 x + 49 = 0$

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 $a \cdot c = 25 \cdot 49 = 1225$ and whose sum is $b = - 74$ .

Factor $- 74$ out of $- 74 x$ .

$25 x^{2} - 74 x + 49 = 0$

Rewrite $- 74$ as $- 25$ plus $- 49$

$25 x^{2} + (- 25 - 49) x + 49 = 0$

Apply the distributive property.

$25 x^{2} - 25 x - 49 x + 49 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(25 x^{2} - 25 x) - 49 x + 49 = 0$

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

$25 x (x - 1) - 49 (x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$(x - 1) (25 x - 49) = 0$

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

$x - 1 = 0$

$25 x - 49 = 0$

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.

$x = 1$

Set $25 x - 49$ equal to $0$ and solve for $x$ .

Set $25 x - 49$ equal to $0$ .

$25 x - 49 = 0$

Solve $25 x - 49 = 0$ for $x$ .

Add $49$ to both sides of the equation.

$25 x = 49$

Divide each term in $25 x = 49$ by $25$ and simplify.

Divide each term in $25 x = 49$ by $25$ .

$\frac{25 x}{25} = \frac{49}{25}$

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{25 x}{25} = \frac{49}{25}$

Divide $x$ by $1$ .

$x = \frac{49}{25}$

The final solution is all the values that make $(x - 1) (25 x - 49) = 0$ true.

$x = 1, \frac{49}{25}$

Exclude the solutions that do not make $\sqrt{x} + 1 = \sqrt{6 x - 6}$ true.

$x = \frac{49}{25}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{49}{25}$

Decimal Form:

$x = 1.96$

Mixed Number Form:

$x = 1 \frac{24}{25}$

Solve for x square root of x+1 = square root of 6x-6

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