Solve the Rational Equation for x square root of 19x+24=-5 square root of x

$\sqrt{19 x + 24} = - 5 \sqrt{x}$

Step 1

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

${\sqrt{19 x + 24}}^{2} = {(- 5 \sqrt{x})}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19 x + 24}$ as ${(19 x + 24)}^{\frac{1}{2}}$ .

${({(19 x + 24)}^{\frac{1}{2}})}^{2} = {(- 5 \sqrt{x})}^{2}$

Simplify the left side.

Simplify ${({(19 x + 24)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(19 x + 24)}^{\frac{1}{2}})}^{2}$ .

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

${(19 x + 24)}^{\frac{1}{2} \cdot 2} = {(- 5 \sqrt{x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(19 x + 24)}^{\frac{1}{2} \cdot 2} = {(- 5 \sqrt{x})}^{2}$

Rewrite the expression.

${(19 x + 24)}^{1} = {(- 5 \sqrt{x})}^{2}$

Simplify.

$19 x + 24 = {(- 5 \sqrt{x})}^{2}$

Simplify the right side.

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

Simplify the expression.

Apply the product rule to $- 5 \sqrt{x}$ .

$19 x + 24 = {(- 5)}^{2} {\sqrt{x}}^{2}$

Raise $- 5$ to the power of $2$ .

$19 x + 24 = 25 {\sqrt{x}}^{2}$

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

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

$19 x + 24 = 25 {(x^{\frac{1}{2}})}^{2}$

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

$19 x + 24 = 25 x^{\frac{1}{2} \cdot 2}$

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

$19 x + 24 = 25 x^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$19 x + 24 = 25 x^{\frac{2}{2}}$

Rewrite the expression.

$19 x + 24 = 25 x^{1}$

Simplify.

$19 x + 24 = 25 x$

Step 3

Solve for $x$ .

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

Subtract $25 x$ from both sides of the equation.

$19 x + 24 - 25 x = 0$

Subtract $25 x$ from $19 x$ .

$- 6 x + 24 = 0$

Subtract $24$ from both sides of the equation.

$- 6 x = - 24$

Divide each term in $- 6 x = - 24$ by $- 6$ and simplify.

Divide each term in $- 6 x = - 24$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 24}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 24}{- 6}$

Divide $x$ by $1$ .

$x = \frac{- 24}{- 6}$

Simplify the right side.

Divide $- 24$ by $- 6$ .

$x = 4$

Step 4

Exclude the solutions that do not make $\sqrt{19 x + 24} = - 5 \sqrt{x}$ true.

$No solution$

Solve the Rational Equation for x square root of 19x+24=-5 square root of x

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