Solve the Rational Equation for x square root of x+19- square root of x+28=-1

$\sqrt{x + 19} - \sqrt{x + 28} = - 1$

Add $\sqrt{x + 28}$ to both sides of the equation.

$\sqrt{x + 19} = - 1 + \sqrt{x + 28}$

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

${\sqrt{x + 19}}^{2} = {(- 1 + \sqrt{x + 28})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 19}$ as ${(x + 19)}^{\frac{1}{2}}$ .

${({(x + 19)}^{\frac{1}{2}})}^{2} = {(- 1 + \sqrt{x + 28})}^{2}$

Simplify the left side.

Simplify ${({(x + 19)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 19)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 19)}^{\frac{1}{2} \cdot 2} = {(- 1 + \sqrt{x + 28})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 19)}^{\frac{1}{2} \cdot 2} = {(- 1 + \sqrt{x + 28})}^{2}$

Rewrite the expression.

${(x + 19)}^{1} = {(- 1 + \sqrt{x + 28})}^{2}$

Simplify.

$x + 19 = {(- 1 + \sqrt{x + 28})}^{2}$

Simplify the right side.

Simplify ${(- 1 + \sqrt{x + 28})}^{2}$ .

Rewrite ${(- 1 + \sqrt{x + 28})}^{2}$ as $(- 1 + \sqrt{x + 28}) (- 1 + \sqrt{x + 28})$ .

$x + 19 = (- 1 + \sqrt{x + 28}) (- 1 + \sqrt{x + 28})$

Expand $(- 1 + \sqrt{x + 28}) (- 1 + \sqrt{x + 28})$ using the FOIL Method.

Apply the distributive property.

$x + 19 = - 1 (- 1 + \sqrt{x + 28}) + \sqrt{x + 28} (- 1 + \sqrt{x + 28})$

Apply the distributive property.

$x + 19 = - 1 \cdot - 1 - 1 \sqrt{x + 28} + \sqrt{x + 28} (- 1 + \sqrt{x + 28})$

Apply the distributive property.

$x + 19 = - 1 \cdot - 1 - 1 \sqrt{x + 28} + \sqrt{x + 28} \cdot - 1 + \sqrt{x + 28} \sqrt{x + 28}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $- 1$ .

$x + 19 = 1 - 1 \sqrt{x + 28} + \sqrt{x + 28} \cdot - 1 + \sqrt{x + 28} \sqrt{x + 28}$

Rewrite $- 1 \sqrt{x + 28}$ as $- \sqrt{x + 28}$ .

$x + 19 = 1 - \sqrt{x + 28} + \sqrt{x + 28} \cdot - 1 + \sqrt{x + 28} \sqrt{x + 28}$

Move $- 1$ to the left of $\sqrt{x + 28}$ .

$x + 19 = 1 - \sqrt{x + 28} - 1 \cdot \sqrt{x + 28} + \sqrt{x + 28} \sqrt{x + 28}$

Rewrite $- 1 \sqrt{x + 28}$ as $- \sqrt{x + 28}$ .

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + \sqrt{x + 28} \sqrt{x + 28}$

Multiply $\sqrt{x + 28} \sqrt{x + 28}$ .

Raise $\sqrt{x + 28}$ to the power of $1$ .

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + {\sqrt{x + 28}}^{1} \sqrt{x + 28}$

Raise $\sqrt{x + 28}$ to the power of $1$ .

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + {\sqrt{x + 28}}^{1} {\sqrt{x + 28}}^{1}$

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

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + {\sqrt{x + 28}}^{1 + 1}$

Add $1$ and $1$ .

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + {\sqrt{x + 28}}^{2}$

Rewrite ${\sqrt{x + 28}}^{2}$ as $x + 28$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 28}$ as ${(x + 28)}^{\frac{1}{2}}$ .

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + {({(x + 28)}^{\frac{1}{2}})}^{2}$

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

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + {(x + 28)}^{\frac{1}{2} \cdot 2}$

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

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + {(x + 28)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + {(x + 28)}^{\frac{2}{2}}$

Rewrite the expression.

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + {(x + 28)}^{1}$

Simplify.

$x + 19 = 1 - \sqrt{x + 28} - \sqrt{x + 28} + x + 28$

Add $1$ and $28$ .

$x + 19 = 29 - \sqrt{x + 28} - \sqrt{x + 28} + x$

Subtract $\sqrt{x + 28}$ from $- \sqrt{x + 28}$ .

$x + 19 = 29 - 2 \sqrt{x + 28} + x$

Solve for $- 2 \sqrt{x + 28}$ .

Rewrite the equation as $29 - 2 \sqrt{x + 28} + x = x + 19$ .

$29 - 2 \sqrt{x + 28} + x = x + 19$

Move all terms not containing $- 2 \sqrt{x + 28}$ to the right side of the equation.

Subtract $29$ from both sides of the equation.

$- 2 \sqrt{x + 28} + x = x + 19 - 29$

Subtract $x$ from both sides of the equation.

$- 2 \sqrt{x + 28} = x + 19 - 29 - x$

Combine the opposite terms in $x + 19 - 29 - x$ .

Subtract $x$ from $x$ .

$- 2 \sqrt{x + 28} = 0 + 19 - 29$

Add $0$ and $19$ .

$- 2 \sqrt{x + 28} = 19 - 29$

Subtract $29$ from $19$ .

$- 2 \sqrt{x + 28} = - 10$

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

${(- 2 \sqrt{x + 28})}^{2} = {(- 10)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 28}$ as ${(x + 28)}^{\frac{1}{2}}$ .

${(- 2 {(x + 28)}^{\frac{1}{2}})}^{2} = {(- 10)}^{2}$

Simplify the left side.

Simplify ${(- 2 {(x + 28)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 2 {(x + 28)}^{\frac{1}{2}}$ .

${(- 2)}^{2} {({(x + 28)}^{\frac{1}{2}})}^{2} = {(- 10)}^{2}$

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

$4 {({(x + 28)}^{\frac{1}{2}})}^{2} = {(- 10)}^{2}$

Multiply the exponents in ${({(x + 28)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(x + 28)}^{\frac{1}{2} \cdot 2} = {(- 10)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(x + 28)}^{\frac{1}{2} \cdot 2} = {(- 10)}^{2}$

Rewrite the expression.

$4 {(x + 28)}^{1} = {(- 10)}^{2}$

Simplify.

$4 (x + 28) = {(- 10)}^{2}$

Apply the distributive property.

$4 x + 4 \cdot 28 = {(- 10)}^{2}$

Multiply $4$ by $28$ .

$4 x + 112 = {(- 10)}^{2}$

Simplify the right side.

Raise $- 10$ to the power of $2$ .

$4 x + 112 = 100$

Solve for $x$ .

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

Subtract $112$ from both sides of the equation.

$4 x = 100 - 112$

Subtract $112$ from $100$ .

$4 x = - 12$

Divide each term in $4 x = - 12$ by $4$ and simplify.

Divide each term in $4 x = - 12$ by $4$ .

$\frac{4 x}{4} = \frac{- 12}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 12}{4}$

Divide $x$ by $1$ .

$x = \frac{- 12}{4}$

Simplify the right side.

Divide $- 12$ by $4$ .

$x = - 3$

Solve the Rational Equation for x square root of x+19- square root of x+28=-1

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