Solve by Factoring x- square root of 3-2x=-6

$x - \sqrt{3 - 2 x} = - 6$

Step 1

Add $6$ to both sides of the equation.

$x - \sqrt{3 - 2 x} + 6 = 0$

Step 2

Move all terms not containing $- \sqrt{3 - 2 x}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- \sqrt{3 - 2 x} + 6 = - x$

Subtract $6$ from both sides of the equation.

$- \sqrt{3 - 2 x} = - x - 6$

Step 3

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

${(- \sqrt{3 - 2 x})}^{2} = {(- x - 6)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - 2 x}$ as ${(3 - 2 x)}^{\frac{1}{2}}$ .

${(- {(3 - 2 x)}^{\frac{1}{2}})}^{2} = {(- x - 6)}^{2}$

Simplify the left side.

Simplify ${(- {(3 - 2 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(3 - 2 x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(3 - 2 x)}^{\frac{1}{2}})}^{2} = {(- x - 6)}^{2}$

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

$1 {({(3 - 2 x)}^{\frac{1}{2}})}^{2} = {(- x - 6)}^{2}$

Multiply ${({(3 - 2 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(3 - 2 x)}^{\frac{1}{2}})}^{2} = {(- x - 6)}^{2}$

Multiply the exponents in ${({(3 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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

${(3 - 2 x)}^{\frac{1}{2} \cdot 2} = {(- x - 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 - 2 x)}^{\frac{1}{2} \cdot 2} = {(- x - 6)}^{2}$

Rewrite the expression.

${(3 - 2 x)}^{1} = {(- x - 6)}^{2}$

Simplify.

$3 - 2 x = {(- x - 6)}^{2}$

Simplify the right side.

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

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

$3 - 2 x = (- x - 6) (- x - 6)$

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

Apply the distributive property.

$3 - 2 x = - x (- x - 6) - 6 (- x - 6)$

Apply the distributive property.

$3 - 2 x = - x (- x) - x \cdot - 6 - 6 (- x - 6)$

Apply the distributive property.

$3 - 2 x = - x (- x) - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 - 2 x = - 1 \cdot - 1 x \cdot x - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

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

Move $x$ .

$3 - 2 x = - 1 \cdot - 1 (x \cdot x) - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

Multiply $x$ by $x$ .

$3 - 2 x = - 1 \cdot - 1 x^{2} - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

Multiply $- 1$ by $- 1$ .

$3 - 2 x = 1 x^{2} - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

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

$3 - 2 x = x^{2} - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

Multiply $- 6$ by $- 1$ .

$3 - 2 x = x^{2} + 6 x - 6 (- x) - 6 \cdot - 6$

Multiply $- 1$ by $- 6$ .

$3 - 2 x = x^{2} + 6 x + 6 x - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$3 - 2 x = x^{2} + 6 x + 6 x + 36$

Add $6 x$ and $6 x$ .

$3 - 2 x = x^{2} + 12 x + 36$

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.

$x^{2} + 12 x + 36 = 3 - 2 x$

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

Add $2 x$ to both sides of the equation.

$x^{2} + 12 x + 36 + 2 x = 3$

Add $12 x$ and $2 x$ .

$x^{2} + 14 x + 36 = 3$

Subtract $3$ from both sides of the equation.

$x^{2} + 14 x + 36 - 3 = 0$

Subtract $3$ from $36$ .

$x^{2} + 14 x + 33 = 0$

Factor $x^{2} + 14 x + 33$ 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 $33$ and whose sum is $14$ .

$3, 11$

Write the factored form using these integers.

$(x + 3) (x + 11) = 0$

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

$x + 3 = 0$

$x + 11 = 0$

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 $x + 11$ equal to $0$ and solve for $x$ .

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Subtract $11$ from both sides of the equation.

$x = - 11$

The final solution is all the values that make $(x + 3) (x + 11) = 0$ true.

$x = - 3, - 11$

Step 6

Exclude the solutions that do not make $x - \sqrt{3 - 2 x} + 6 = 0$ true.

$x = - 3$

Solve by Factoring x- square root of 3-2x=-6

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