Solve for x square root of x+9=1+ square root of x

$\sqrt{x + 9} = 1 + \sqrt{x}$

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

${\sqrt{x + 9}}^{2} = {(1 + \sqrt{x})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 9}$ as ${(x + 9)}^{\frac{1}{2}}$ .

${({(x + 9)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{x})}^{2}$

Simplify the left side.

Simplify ${({(x + 9)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 9)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 9)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 9)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{x})}^{2}$

Rewrite the expression.

${(x + 9)}^{1} = {(1 + \sqrt{x})}^{2}$

Simplify.

$x + 9 = {(1 + \sqrt{x})}^{2}$

Simplify the right side.

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

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

$x + 9 = (1 + \sqrt{x}) (1 + \sqrt{x})$

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

Apply the distributive property.

$x + 9 = 1 (1 + \sqrt{x}) + \sqrt{x} (1 + \sqrt{x})$

Apply the distributive property.

$x + 9 = 1 \cdot 1 + 1 \sqrt{x} + \sqrt{x} (1 + \sqrt{x})$

Apply the distributive property.

$x + 9 = 1 \cdot 1 + 1 \sqrt{x} + \sqrt{x} \cdot 1 + \sqrt{x} \sqrt{x}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$x + 9 = 1 + 1 \sqrt{x} + \sqrt{x} \cdot 1 + \sqrt{x} \sqrt{x}$

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

$x + 9 = 1 + \sqrt{x} + \sqrt{x} \cdot 1 + \sqrt{x} \sqrt{x}$

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

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + \sqrt{x} \sqrt{x}$

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

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

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}$

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

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}$

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

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + {\sqrt{x}}^{1 + 1}$

Add $1$ and $1$ .

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + {\sqrt{x}}^{2}$

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

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

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + {(x^{\frac{1}{2}})}^{2}$

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

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + x^{\frac{1}{2} \cdot 2}$

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

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + x^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + x^{\frac{2}{2}}$

Rewrite the expression.

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + x^{1}$

Simplify.

$x + 9 = 1 + \sqrt{x} + \sqrt{x} + x$

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

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

Solve for $2 \sqrt{x}$ .

Rewrite the equation as $1 + 2 \sqrt{x} + x = x + 9$ .

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

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

Subtract $1$ from both sides of the equation.

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

Subtract $x$ from both sides of the equation.

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

Combine the opposite terms in $x + 9 - 1 - x$ .

Subtract $x$ from $x$ .

$2 \sqrt{x} = 0 + 9 - 1$

Add $0$ and $9$ .

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

Subtract $1$ from $9$ .

$2 \sqrt{x} = 8$

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $2 x^{\frac{1}{2}}$ .

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

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

$4 {(x^{\frac{1}{2}})}^{2} = 8^{2}$

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

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

$4 x^{\frac{1}{2} \cdot 2} = 8^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x^{\frac{1}{2} \cdot 2} = 8^{2}$

Rewrite the expression.

$4 x^{1} = 8^{2}$

Simplify.

$4 x = 8^{2}$

Simplify the right side.

Raise $8$ to the power of $2$ .

$4 x = 64$

Divide each term in $4 x = 64$ by $4$ and simplify.

Divide each term in $4 x = 64$ by $4$ .

$\frac{4 x}{4} = \frac{64}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{64}{4}$

Divide $x$ by $1$ .

$x = \frac{64}{4}$

Simplify the right side.

Divide $64$ by $4$ .

$x = 16$

Solve for x square root of x+9=1+ square root of x

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