Solve the Rational Equation for x square root of 2x+15=x+6

$\sqrt{2 x + 15} = x + 6$

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

${\sqrt{2 x + 15}}^{2} = {(x + 6)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 15}$ as ${(2 x + 15)}^{\frac{1}{2}}$ .

${({(2 x + 15)}^{\frac{1}{2}})}^{2} = {(x + 6)}^{2}$

Simplify the left side.

Simplify ${({(2 x + 15)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 x + 15)}^{\frac{1}{2}})}^{2}$ .

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

${(2 x + 15)}^{\frac{1}{2} \cdot 2} = {(x + 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x + 15)}^{\frac{1}{2} \cdot 2} = {(x + 6)}^{2}$

Rewrite the expression.

${(2 x + 15)}^{1} = {(x + 6)}^{2}$

Simplify.

$2 x + 15 = {(x + 6)}^{2}$

Simplify the right side.

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

Rewrite ${(x + 6)}^{2}$ as $(x + 6) (x + 6)$ .

$2 x + 15 = (x + 6) (x + 6)$

Expand $(x + 6) (x + 6)$ using the FOIL Method.

Apply the distributive property.

$2 x + 15 = x (x + 6) + 6 (x + 6)$

Apply the distributive property.

$2 x + 15 = x \cdot x + x \cdot 6 + 6 (x + 6)$

Apply the distributive property.

$2 x + 15 = x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$2 x + 15 = x^{2} + x \cdot 6 + 6 x + 6 \cdot 6$

Move $6$ to the left of $x$ .

$2 x + 15 = x^{2} + 6 \cdot x + 6 x + 6 \cdot 6$

Multiply $6$ by $6$ .

$2 x + 15 = x^{2} + 6 x + 6 x + 36$

Add $6 x$ and $6 x$ .

$2 x + 15 = x^{2} + 12 x + 36$

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.

$x^{2} + 12 x + 36 = 2 x + 15$

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

Subtract $2 x$ from both sides of the equation.

$x^{2} + 12 x + 36 - 2 x = 15$

Subtract $2 x$ from $12 x$ .

$x^{2} + 10 x + 36 = 15$

Subtract $15$ from both sides of the equation.

$x^{2} + 10 x + 36 - 15 = 0$

Subtract $15$ from $36$ .

$x^{2} + 10 x + 21 = 0$

Factor $x^{2} + 10 x + 21$ 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 $21$ and whose sum is $10$ .

$3, 7$

Write the factored form using these integers.

$(x + 3) (x + 7) = 0$

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

$x + 3 = 0$

$x + 7 = 0$

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 $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

The final solution is all the values that make $(x + 3) (x + 7) = 0$ true.

$x = - 3, - 7$

Exclude the solutions that do not make $\sqrt{2 x + 15} = x + 6$ true.

$x = - 3$

Solve the Rational Equation for x square root of 2x+15=x+6

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