Solve the Rational Equation for x square root of 2x+1+7=x

$\sqrt{2 x + 1} + 7 = x$

Step 1

Subtract $7$ from both sides of the equation.

$\sqrt{2 x + 1} = x - 7$

Step 2

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

${\sqrt{2 x + 1}}^{2} = {(x - 7)}^{2}$

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(x - 7)}^{2}$ .

Rewrite ${(x - 7)}^{2}$ as $(x - 7) (x - 7)$ .

$2 x + 1 = (x - 7) (x - 7)$

Expand $(x - 7) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$2 x + 1 = x (x - 7) - 7 (x - 7)$

Apply the distributive property.

$2 x + 1 = x \cdot x + x \cdot - 7 - 7 (x - 7)$

Apply the distributive property.

$2 x + 1 = x \cdot x + x \cdot - 7 - 7 x - 7 \cdot - 7$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$2 x + 1 = x^{2} + x \cdot - 7 - 7 x - 7 \cdot - 7$

Move $- 7$ to the left of $x$ .

$2 x + 1 = x^{2} - 7 \cdot x - 7 x - 7 \cdot - 7$

Multiply $- 7$ by $- 7$ .

$2 x + 1 = x^{2} - 7 x - 7 x + 49$

Subtract $7 x$ from $- 7 x$ .

$2 x + 1 = x^{2} - 14 x + 49$

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.

$x^{2} - 14 x + 49 = 2 x + 1$

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

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

$x^{2} - 14 x + 49 - 2 x = 1$

Subtract $2 x$ from $- 14 x$ .

$x^{2} - 16 x + 49 = 1$

Subtract $1$ from both sides of the equation.

$x^{2} - 16 x + 49 - 1 = 0$

Subtract $1$ from $49$ .

$x^{2} - 16 x + 48 = 0$

Factor $x^{2} - 16 x + 48$ 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 $48$ and whose sum is $- 16$ .

$- 12, - 4$

Write the factored form using these integers.

$(x - 12) (x - 4) = 0$

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

$x - 12 = 0$

$x - 4 = 0$

Set $x - 12$ equal to $0$ and solve for $x$ .

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Add $12$ to both sides of the equation.

$x = 12$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(x - 12) (x - 4) = 0$ true.

$x = 12, 4$

Step 5

Exclude the solutions that do not make $\sqrt{2 x + 1} + 7 = x$ true.

$x = 12$

Solve the Rational Equation for x square root of 2x+1+7=x

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