Solve the Rational Equation for x square root of x+6-4=x

$\sqrt{x + 6} - 4 = x$

Step 1

Add $4$ to both sides of the equation.

$\sqrt{x + 6} = x + 4$

Step 2

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

${\sqrt{x + 6}}^{2} = {(x + 4)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 6}$ as ${(x + 6)}^{\frac{1}{2}}$ .

${({(x + 6)}^{\frac{1}{2}})}^{2} = {(x + 4)}^{2}$

Simplify the left side.

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

Multiply the exponents in ${({(x + 6)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 6)}^{\frac{1}{2} \cdot 2} = {(x + 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 6)}^{\frac{1}{2} \cdot 2} = {(x + 4)}^{2}$

Rewrite the expression.

${(x + 6)}^{1} = {(x + 4)}^{2}$

Simplify.

$x + 6 = {(x + 4)}^{2}$

Simplify the right side.

Simplify ${(x + 4)}^{2}$ .

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$x + 6 = (x + 4) (x + 4)$

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$x + 6 = x (x + 4) + 4 (x + 4)$

Apply the distributive property.

$x + 6 = x \cdot x + x \cdot 4 + 4 (x + 4)$

Apply the distributive property.

$x + 6 = x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x + 6 = x^{2} + x \cdot 4 + 4 x + 4 \cdot 4$

Move $4$ to the left of $x$ .

$x + 6 = x^{2} + 4 \cdot x + 4 x + 4 \cdot 4$

Multiply $4$ by $4$ .

$x + 6 = x^{2} + 4 x + 4 x + 16$

Add $4 x$ and $4 x$ .

$x + 6 = x^{2} + 8 x + 16$

Step 4

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.

$x^{2} + 8 x + 16 = x + 6$

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

Subtract $x$ from both sides of the equation.

$x^{2} + 8 x + 16 - x = 6$

Subtract $x$ from $8 x$ .

$x^{2} + 7 x + 16 = 6$

Subtract $6$ from both sides of the equation.

$x^{2} + 7 x + 16 - 6 = 0$

Subtract $6$ from $16$ .

$x^{2} + 7 x + 10 = 0$

Factor $x^{2} + 7 x + 10$ 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 $10$ and whose sum is $7$ .

$2, 5$

Write the factored form using these integers.

$(x + 2) (x + 5) = 0$

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

$x + 2 = 0$

$x + 5 = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $(x + 2) (x + 5) = 0$ true.

$x = - 2, - 5$

Step 5

Exclude the solutions that do not make $\sqrt{x + 6} - 4 = x$ true.

$x = - 2$

Solve the Rational Equation for x square root of x+6-4=x

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