Solve the Rational Equation for x x+3 = square root of x+33

$x + 3 = \sqrt{x + 33}$

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{x + 33} = x + 3$

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

${\sqrt{x + 33}}^{2} = {(x + 3)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 33}$ as ${(x + 33)}^{\frac{1}{2}}$ .

${({(x + 33)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Simplify the left side.

Simplify ${({(x + 33)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 33)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 33)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 33)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Rewrite the expression.

${(x + 33)}^{1} = {(x + 3)}^{2}$

Simplify.

$x + 33 = {(x + 3)}^{2}$

Simplify the right side.

Simplify ${(x + 3)}^{2}$ .

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$x + 33 = (x + 3) (x + 3)$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$x + 33 = x (x + 3) + 3 (x + 3)$

Apply the distributive property.

$x + 33 = x \cdot x + x \cdot 3 + 3 (x + 3)$

Apply the distributive property.

$x + 33 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x + 33 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Move $3$ to the left of $x$ .

$x + 33 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$x + 33 = x^{2} + 3 x + 3 x + 9$

Add $3 x$ and $3 x$ .

$x + 33 = x^{2} + 6 x + 9$

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 6 x + 9 = x + 33$

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

Subtract $x$ from both sides of the equation.

$x^{2} + 6 x + 9 - x = 33$

Subtract $x$ from $6 x$ .

$x^{2} + 5 x + 9 = 33$

Subtract $33$ from both sides of the equation.

$x^{2} + 5 x + 9 - 33 = 0$

Subtract $33$ from $9$ .

$x^{2} + 5 x - 24 = 0$

Factor $x^{2} + 5 x - 24$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $5$ .

$- 3, 8$

Write the factored form using these integers.

$(x - 3) (x + 8) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x + 8 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $(x - 3) (x + 8) = 0$ true.

$x = 3, - 8$

Exclude the solutions that do not make $x + 3 = \sqrt{x + 33}$ true.

$x = 3$

Solve the Rational Equation for x x+3 = square root of x+33

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