Solve by Factoring x+1 = square root of 7x+15

$x + 1 = \sqrt{7 x + 15}$

Step 1

Subtract $\sqrt{7 x + 15}$ from both sides of the equation.

$x + 1 - \sqrt{7 x + 15} = 0$

Step 2

Move all terms not containing $- \sqrt{7 x + 15}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$1 - \sqrt{7 x + 15} = - x$

Subtract $1$ from both sides of the equation.

$- \sqrt{7 x + 15} = - x - 1$

Step 3

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

${(- \sqrt{7 x + 15})}^{2} = {(- x - 1)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x + 15}$ as ${(7 x + 15)}^{\frac{1}{2}}$ .

${(- {(7 x + 15)}^{\frac{1}{2}})}^{2} = {(- x - 1)}^{2}$

Simplify the left side.

Simplify ${(- {(7 x + 15)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(7 x + 15)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(7 x + 15)}^{\frac{1}{2}})}^{2} = {(- x - 1)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(7 x + 15)}^{\frac{1}{2}})}^{2} = {(- x - 1)}^{2}$

Multiply ${({(7 x + 15)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(7 x + 15)}^{\frac{1}{2}})}^{2} = {(- x - 1)}^{2}$

Multiply the exponents in ${({(7 x + 15)}^{\frac{1}{2}})}^{2}$ .

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

${(7 x + 15)}^{\frac{1}{2} \cdot 2} = {(- x - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(7 x + 15)}^{\frac{1}{2} \cdot 2} = {(- x - 1)}^{2}$

Rewrite the expression.

${(7 x + 15)}^{1} = {(- x - 1)}^{2}$

Simplify.

$7 x + 15 = {(- x - 1)}^{2}$

Simplify the right side.

Simplify ${(- x - 1)}^{2}$ .

Rewrite ${(- x - 1)}^{2}$ as $(- x - 1) (- x - 1)$ .

$7 x + 15 = (- x - 1) (- x - 1)$

Expand $(- x - 1) (- x - 1)$ using the FOIL Method.

Apply the distributive property.

$7 x + 15 = - x (- x - 1) - 1 (- x - 1)$

Apply the distributive property.

$7 x + 15 = - x (- x) - x \cdot - 1 - 1 (- x - 1)$

Apply the distributive property.

$7 x + 15 = - x (- x) - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$7 x + 15 = - 1 \cdot - 1 x \cdot x - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$7 x + 15 = - 1 \cdot - 1 (x \cdot x) - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Multiply $x$ by $x$ .

$7 x + 15 = - 1 \cdot - 1 x^{2} - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$7 x + 15 = 1 x^{2} - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Multiply $x^{2}$ by $1$ .

$7 x + 15 = x^{2} - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Multiply $- x \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$7 x + 15 = x^{2} + 1 x - 1 (- x) - 1 \cdot - 1$

Multiply $x$ by $1$ .

$7 x + 15 = x^{2} + x - 1 (- x) - 1 \cdot - 1$

Multiply $- 1 (- x)$ .

Multiply $- 1$ by $- 1$ .

$7 x + 15 = x^{2} + x + 1 x - 1 \cdot - 1$

Multiply $x$ by $1$ .

$7 x + 15 = x^{2} + x + x - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$7 x + 15 = x^{2} + x + x + 1$

Add $x$ and $x$ .

$7 x + 15 = x^{2} + 2 x + 1$

Step 5

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.

$x^{2} + 2 x + 1 = 7 x + 15$

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

Subtract $7 x$ from both sides of the equation.

$x^{2} + 2 x + 1 - 7 x = 15$

Subtract $7 x$ from $2 x$ .

$x^{2} - 5 x + 1 = 15$

Subtract $15$ from both sides of the equation.

$x^{2} - 5 x + 1 - 15 = 0$

Subtract $15$ from $1$ .

$x^{2} - 5 x - 14 = 0$

Factor $x^{2} - 5 x - 14$ 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 $- 14$ and whose sum is $- 5$ .

$- 7, 2$

Write the factored form using these integers.

$(x - 7) (x + 2) = 0$

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

$x - 7 = 0$

$x + 2 = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

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

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

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.

$x = - 2$

The final solution is all the values that make $(x - 7) (x + 2) = 0$ true.

$x = 7, - 2$

Step 6

Exclude the solutions that do not make $x + 1 - \sqrt{7 x + 15} = 0$ true.

$x = 7$

Solve by Factoring x+1 = square root of 7x+15

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