Solve the Rational Equation for x square root of x+7-5=x

$\sqrt{x + 7} - 5 = x$

Step 1

Add $5$ to both sides of the equation.

$\sqrt{x + 7} = x + 5$

Step 2

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

${\sqrt{x + 7}}^{2} = {(x + 5)}^{2}$

Step 3

Simplify each side of the equation.

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

${({(x + 7)}^{\frac{1}{2}})}^{2} = {(x + 5)}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(x + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(x + 5)}^{2}$

Rewrite the expression.

${(x + 7)}^{1} = {(x + 5)}^{2}$

Simplify.

$x + 7 = {(x + 5)}^{2}$

Simplify the right side.

Simplify ${(x + 5)}^{2}$ .

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$x + 7 = (x + 5) (x + 5)$

Expand $(x + 5) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$x + 7 = x (x + 5) + 5 (x + 5)$

Apply the distributive property.

$x + 7 = x \cdot x + x \cdot 5 + 5 (x + 5)$

Apply the distributive property.

$x + 7 = x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x + 7 = x^{2} + x \cdot 5 + 5 x + 5 \cdot 5$

Move $5$ to the left of $x$ .

$x + 7 = x^{2} + 5 \cdot x + 5 x + 5 \cdot 5$

Multiply $5$ by $5$ .

$x + 7 = x^{2} + 5 x + 5 x + 25$

Add $5 x$ and $5 x$ .

$x + 7 = x^{2} + 10 x + 25$

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.

$x^{2} + 10 x + 25 = x + 7$

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

Subtract $x$ from both sides of the equation.

$x^{2} + 10 x + 25 - x = 7$

Subtract $x$ from $10 x$ .

$x^{2} + 9 x + 25 = 7$

Subtract $7$ from both sides of the equation.

$x^{2} + 9 x + 25 - 7 = 0$

Subtract $7$ from $25$ .

$x^{2} + 9 x + 18 = 0$

Factor $x^{2} + 9 x + 18$ 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 $18$ and whose sum is $9$ .

$3, 6$

Write the factored form using these integers.

$(x + 3) (x + 6) = 0$

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

$x + 3 = 0$

$x + 6 = 0$

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x + 3) (x + 6) = 0$ true.

$x = - 3, - 6$

Step 5

Exclude the solutions that do not make $\sqrt{x + 7} - 5 = x$ true.

$x = - 3$

Solve the Rational Equation for x square root of x+7-5=x

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