Solve for x square root of x-1 = square root of x-4

$\sqrt{x} - 1 = \sqrt{x - 4}$

Add $1$ to both sides of the equation.

$\sqrt{x} = \sqrt{x - 4} + 1$

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

${\sqrt{x}}^{2} = {(\sqrt{x - 4} + 1)}^{2}$

Simplify each side of the equation.

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x^{1} = {(\sqrt{x - 4} + 1)}^{2}$

Simplify.

$x = {(\sqrt{x - 4} + 1)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x - 4} + 1)}^{2}$ .

Rewrite ${(\sqrt{x - 4} + 1)}^{2}$ as $(\sqrt{x - 4} + 1) (\sqrt{x - 4} + 1)$ .

$x = (\sqrt{x - 4} + 1) (\sqrt{x - 4} + 1)$

Expand $(\sqrt{x - 4} + 1) (\sqrt{x - 4} + 1)$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{x - 4} (\sqrt{x - 4} + 1) + 1 (\sqrt{x - 4} + 1)$

Apply the distributive property.

$x = \sqrt{x - 4} \sqrt{x - 4} + \sqrt{x - 4} \cdot 1 + 1 (\sqrt{x - 4} + 1)$

Apply the distributive property.

$x = \sqrt{x - 4} \sqrt{x - 4} + \sqrt{x - 4} \cdot 1 + 1 \sqrt{x - 4} + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x - 4} \sqrt{x - 4}$ .

Raise $\sqrt{x - 4}$ to the power of $1$ .

$x = {\sqrt{x - 4}}^{1} \sqrt{x - 4} + \sqrt{x - 4} \cdot 1 + 1 \sqrt{x - 4} + 1 \cdot 1$

Raise $\sqrt{x - 4}$ to the power of $1$ .

$x = {\sqrt{x - 4}}^{1} {\sqrt{x - 4}}^{1} + \sqrt{x - 4} \cdot 1 + 1 \sqrt{x - 4} + 1 \cdot 1$

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

$x = {\sqrt{x - 4}}^{1 + 1} + \sqrt{x - 4} \cdot 1 + 1 \sqrt{x - 4} + 1 \cdot 1$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x - 4}}^{2}$ as $x - 4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = {(x - 4)}^{1} + \sqrt{x - 4} \cdot 1 + 1 \sqrt{x - 4} + 1 \cdot 1$

Simplify.

$x = x - 4 + \sqrt{x - 4} \cdot 1 + 1 \sqrt{x - 4} + 1 \cdot 1$

Multiply $\sqrt{x - 4}$ by $1$ .

$x = x - 4 + \sqrt{x - 4} + 1 \sqrt{x - 4} + 1 \cdot 1$

Multiply $\sqrt{x - 4}$ by $1$ .

$x = x - 4 + \sqrt{x - 4} + \sqrt{x - 4} + 1 \cdot 1$

Multiply $1$ by $1$ .

$x = x - 4 + \sqrt{x - 4} + \sqrt{x - 4} + 1$

Add $- 4$ and $1$ .

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

Add $\sqrt{x - 4}$ and $\sqrt{x - 4}$ .

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

Solve for $2 \sqrt{x - 4}$ .

Rewrite the equation as $x - 3 + 2 \sqrt{x - 4} = x$ .

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

Move all terms not containing $2 \sqrt{x - 4}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

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

Add $3$ to both sides of the equation.

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

Subtract $x$ from $x$ .

$2 \sqrt{x - 4} = 0 + 3$

Add $0$ and $3$ .

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

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

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

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

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

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

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

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

Multiply the exponents in ${({(x - 4)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $4$ by $- 4$ .

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

Simplify the right side.

Raise $3$ to the power of $2$ .

$4 x - 16 = 9$

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Add $16$ to both sides of the equation.

$4 x = 9 + 16$

Add $9$ and $16$ .

$4 x = 25$

Divide each term in $4 x = 25$ by $4$ and simplify.

Divide each term in $4 x = 25$ by $4$ .

$\frac{4 x}{4} = \frac{25}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{25}{4}$

Divide $x$ by $1$ .

$x = \frac{25}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{25}{4}$

Decimal Form:

$x = 6.25$

Mixed Number Form:

$x = 6 \frac{1}{4}$

Solve for x square root of x-1 = square root of x-4

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