Solve by Factoring x- square root of 7-6x=-14

$x - \sqrt{7 - 6 x} = - 14$

Step 1

Add $14$ to both sides of the equation.

$x - \sqrt{7 - 6 x} + 14 = 0$

Step 2

Move all terms not containing $- \sqrt{7 - 6 x}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- \sqrt{7 - 6 x} + 14 = - x$

Subtract $14$ from both sides of the equation.

$- \sqrt{7 - 6 x} = - x - 14$

Step 3

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

${(- \sqrt{7 - 6 x})}^{2} = {(- x - 14)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 - 6 x}$ as ${(7 - 6 x)}^{\frac{1}{2}}$ .

${(- {(7 - 6 x)}^{\frac{1}{2}})}^{2} = {(- x - 14)}^{2}$

Simplify the left side.

Simplify ${(- {(7 - 6 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(7 - 6 x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(7 - 6 x)}^{\frac{1}{2}})}^{2} = {(- x - 14)}^{2}$

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

$1 {({(7 - 6 x)}^{\frac{1}{2}})}^{2} = {(- x - 14)}^{2}$

Multiply ${({(7 - 6 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(7 - 6 x)}^{\frac{1}{2}})}^{2} = {(- x - 14)}^{2}$

Multiply the exponents in ${({(7 - 6 x)}^{\frac{1}{2}})}^{2}$ .

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

${(7 - 6 x)}^{\frac{1}{2} \cdot 2} = {(- x - 14)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(7 - 6 x)}^{\frac{1}{2} \cdot 2} = {(- x - 14)}^{2}$

Rewrite the expression.

${(7 - 6 x)}^{1} = {(- x - 14)}^{2}$

Simplify.

$7 - 6 x = {(- x - 14)}^{2}$

Simplify the right side.

Simplify ${(- x - 14)}^{2}$ .

Rewrite ${(- x - 14)}^{2}$ as $(- x - 14) (- x - 14)$ .

$7 - 6 x = (- x - 14) (- x - 14)$

Expand $(- x - 14) (- x - 14)$ using the FOIL Method.

Apply the distributive property.

$7 - 6 x = - x (- x - 14) - 14 (- x - 14)$

Apply the distributive property.

$7 - 6 x = - x (- x) - x \cdot - 14 - 14 (- x - 14)$

Apply the distributive property.

$7 - 6 x = - x (- x) - x \cdot - 14 - 14 (- x) - 14 \cdot - 14$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$7 - 6 x = - 1 \cdot - 1 x \cdot x - x \cdot - 14 - 14 (- x) - 14 \cdot - 14$

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

Move $x$ .

$7 - 6 x = - 1 \cdot - 1 (x \cdot x) - x \cdot - 14 - 14 (- x) - 14 \cdot - 14$

Multiply $x$ by $x$ .

$7 - 6 x = - 1 \cdot - 1 x^{2} - x \cdot - 14 - 14 (- x) - 14 \cdot - 14$

Multiply $- 1$ by $- 1$ .

$7 - 6 x = 1 x^{2} - x \cdot - 14 - 14 (- x) - 14 \cdot - 14$

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

$7 - 6 x = x^{2} - x \cdot - 14 - 14 (- x) - 14 \cdot - 14$

Multiply $- 14$ by $- 1$ .

$7 - 6 x = x^{2} + 14 x - 14 (- x) - 14 \cdot - 14$

Multiply $- 1$ by $- 14$ .

$7 - 6 x = x^{2} + 14 x + 14 x - 14 \cdot - 14$

Multiply $- 14$ by $- 14$ .

$7 - 6 x = x^{2} + 14 x + 14 x + 196$

Add $14 x$ and $14 x$ .

$7 - 6 x = x^{2} + 28 x + 196$

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.

$x^{2} + 28 x + 196 = 7 - 6 x$

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

Add $6 x$ to both sides of the equation.

$x^{2} + 28 x + 196 + 6 x = 7$

Add $28 x$ and $6 x$ .

$x^{2} + 34 x + 196 = 7$

Subtract $7$ from both sides of the equation.

$x^{2} + 34 x + 196 - 7 = 0$

Subtract $7$ from $196$ .

$x^{2} + 34 x + 189 = 0$

Factor $x^{2} + 34 x + 189$ 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 $189$ and whose sum is $34$ .

$7, 27$

Write the factored form using these integers.

$(x + 7) (x + 27) = 0$

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

$x + 7 = 0$

$x + 27 = 0$

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.

$x = - 7$

Set $x + 27$ equal to $0$ and solve for $x$ .

Set $x + 27$ equal to $0$ .

$x + 27 = 0$

Subtract $27$ from both sides of the equation.

$x = - 27$

The final solution is all the values that make $(x + 7) (x + 27) = 0$ true.

$x = - 7, - 27$

Step 6

Exclude the solutions that do not make $x - \sqrt{7 - 6 x} + 14 = 0$ true.

$x = - 7$

Solve by Factoring x- square root of 7-6x=-14

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