Solve for x square root of x+7+ square root of x=3

$\sqrt{x + 7} + \sqrt{x} = 3$

Step 1

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

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

Step 2

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

${\sqrt{x + 7}}^{2} = {(3 - \sqrt{x})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

${({(x + 7)}^{\frac{1}{2}})}^{2} = {(3 - \sqrt{x})}^{2}$

Simplify the left side.

Simplify ${({(x + 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 7)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(3 - \sqrt{x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(3 - \sqrt{x})}^{2}$

Rewrite the expression.

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

Simplify.

$x + 7 = {(3 - \sqrt{x})}^{2}$

Simplify the right side.

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

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

$x + 7 = (3 - \sqrt{x}) (3 - \sqrt{x})$

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

Apply the distributive property.

$x + 7 = 3 (3 - \sqrt{x}) - \sqrt{x} (3 - \sqrt{x})$

Apply the distributive property.

$x + 7 = 3 \cdot 3 + 3 (- \sqrt{x}) - \sqrt{x} (3 - \sqrt{x})$

Apply the distributive property.

$x + 7 = 3 \cdot 3 + 3 (- \sqrt{x}) - \sqrt{x} \cdot 3 - \sqrt{x} (- \sqrt{x})$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$x + 7 = 9 + 3 (- \sqrt{x}) - \sqrt{x} \cdot 3 - \sqrt{x} (- \sqrt{x})$

Multiply $- 1$ by $3$ .

$x + 7 = 9 - 3 \sqrt{x} - \sqrt{x} \cdot 3 - \sqrt{x} (- \sqrt{x})$

Multiply $3$ by $- 1$ .

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} - \sqrt{x} (- \sqrt{x})$

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

Multiply $- 1$ by $- 1$ .

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + 1 \sqrt{x} \sqrt{x}$

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

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + \sqrt{x} \sqrt{x}$

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

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}$

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

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}$

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

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + {\sqrt{x}}^{1 + 1}$

Add $1$ and $1$ .

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + {\sqrt{x}}^{2}$

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

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

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + {(x^{\frac{1}{2}})}^{2}$

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

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + x^{\frac{1}{2} \cdot 2}$

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

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + x^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + x^{\frac{2}{2}}$

Rewrite the expression.

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + x^{1}$

Simplify.

$x + 7 = 9 - 3 \sqrt{x} - 3 \sqrt{x} + x$

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

$x + 7 = 9 - 6 \sqrt{x} + x$

Step 4

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

Rewrite the equation as $9 - 6 \sqrt{x} + x = x + 7$ .

$9 - 6 \sqrt{x} + x = x + 7$

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

Subtract $9$ from both sides of the equation.

$- 6 \sqrt{x} + x = x + 7 - 9$

Subtract $x$ from both sides of the equation.

$- 6 \sqrt{x} = x + 7 - 9 - x$

Combine the opposite terms in $x + 7 - 9 - x$ .

Subtract $x$ from $x$ .

$- 6 \sqrt{x} = 0 + 7 - 9$

Add $0$ and $7$ .

$- 6 \sqrt{x} = 7 - 9$

Subtract $9$ from $7$ .

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

Step 5

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

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

Step 6

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$36 x = 4$

Step 7

Divide each term in $36 x = 4$ by $36$ and simplify.

Divide each term in $36 x = 4$ by $36$ .

$\frac{36 x}{36} = \frac{4}{36}$

Simplify the left side.

Cancel the common factor of $36$ .

Cancel the common factor.

$\frac{36 x}{36} = \frac{4}{36}$

Divide $x$ by $1$ .

$x = \frac{4}{36}$

Simplify the right side.

Cancel the common factor of $4$ and $36$ .

Factor $4$ out of $4$ .

$x = \frac{4 (1)}{36}$

Cancel the common factors.

Factor $4$ out of $36$ .

$x = \frac{4 \cdot 1}{4 \cdot 9}$

Cancel the common factor.

$x = \frac{4 \cdot 1}{4 \cdot 9}$

Rewrite the expression.

$x = \frac{1}{9}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{9}$

Decimal Form:

$x = 0.\overline{1}$

Solve for x square root of x+7+ square root of x=3

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