Solve by Factoring x-3 = square root of 30-2x

$x - 3 = \sqrt{30 - 2 x}$

Step 1

Subtract $\sqrt{30 - 2 x}$ from both sides of the equation.

$x - 3 - \sqrt{30 - 2 x} = 0$

Step 2

Factor $2$ out of $30 - 2 x$ .

Factor $2$ out of $30$ .

$x - 3 - \sqrt{2 (15) - 2 x} = 0$

Factor $2$ out of $- 2 x$ .

$x - 3 - \sqrt{2 (15) + 2 (- x)} = 0$

Factor $2$ out of $2 (15) + 2 (- x)$ .

$x - 3 - \sqrt{2 (15 - x)} = 0$

Step 3

Move all terms not containing $- \sqrt{2 (15 - x)}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- 3 - \sqrt{2 (15 - x)} = - x$

Add $3$ to both sides of the equation.

$- \sqrt{2 (15 - x)} = - x + 3$

Step 4

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

${(- \sqrt{2 (15 - x)})}^{2} = {(- x + 3)}^{2}$

Step 5

Simplify each side of the equation.

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

${(- {(2 (15 - x))}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

Simplify the left side.

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

Apply the distributive property.

${(- {(2 \cdot 15 + 2 (- x))}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

Simplify the expression.

Multiply $2$ by $15$ .

${(- {(30 + 2 (- x))}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

Multiply $- 1$ by $2$ .

${(- {(30 - 2 x)}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

Apply the product rule to $- {(30 - 2 x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(30 - 2 x)}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

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

$1 {({(30 - 2 x)}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

Multiply ${({(30 - 2 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(30 - 2 x)}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

Multiply the exponents in ${({(30 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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

${(30 - 2 x)}^{\frac{1}{2} \cdot 2} = {(- x + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(30 - 2 x)}^{\frac{1}{2} \cdot 2} = {(- x + 3)}^{2}$

Rewrite the expression.

${(30 - 2 x)}^{1} = {(- x + 3)}^{2}$

Simplify.

$30 - 2 x = {(- x + 3)}^{2}$

Simplify the right side.

Simplify ${(- x + 3)}^{2}$ .

Rewrite ${(- x + 3)}^{2}$ as $(- x + 3) (- x + 3)$ .

$30 - 2 x = (- x + 3) (- x + 3)$

Expand $(- x + 3) (- x + 3)$ using the FOIL Method.

Apply the distributive property.

$30 - 2 x = - x (- x + 3) + 3 (- x + 3)$

Apply the distributive property.

$30 - 2 x = - x (- x) - x \cdot 3 + 3 (- x + 3)$

Apply the distributive property.

$30 - 2 x = - x (- x) - x \cdot 3 + 3 (- x) + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$30 - 2 x = - 1 \cdot - 1 x \cdot x - x \cdot 3 + 3 (- x) + 3 \cdot 3$

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

Move $x$ .

$30 - 2 x = - 1 \cdot - 1 (x \cdot x) - x \cdot 3 + 3 (- x) + 3 \cdot 3$

Multiply $x$ by $x$ .

$30 - 2 x = - 1 \cdot - 1 x^{2} - x \cdot 3 + 3 (- x) + 3 \cdot 3$

Multiply $- 1$ by $- 1$ .

$30 - 2 x = 1 x^{2} - x \cdot 3 + 3 (- x) + 3 \cdot 3$

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

$30 - 2 x = x^{2} - x \cdot 3 + 3 (- x) + 3 \cdot 3$

Multiply $3$ by $- 1$ .

$30 - 2 x = x^{2} - 3 x + 3 (- x) + 3 \cdot 3$

Multiply $- 1$ by $3$ .

$30 - 2 x = x^{2} - 3 x - 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$30 - 2 x = x^{2} - 3 x - 3 x + 9$

Subtract $3 x$ from $- 3 x$ .

$30 - 2 x = x^{2} - 6 x + 9$

Step 6

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.

$x^{2} - 6 x + 9 = 30 - 2 x$

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

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

$x^{2} - 6 x + 9 + 2 x = 30$

Add $- 6 x$ and $2 x$ .

$x^{2} - 4 x + 9 = 30$

Subtract $30$ from both sides of the equation.

$x^{2} - 4 x + 9 - 30 = 0$

Subtract $30$ from $9$ .

$x^{2} - 4 x - 21 = 0$

Factor $x^{2} - 4 x - 21$ 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 $- 4$ .

$- 7, 3$

Write the factored form using these integers.

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

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

$x - 7 = 0$

$x + 3 = 0$

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 $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

$x = 7, - 3$

Step 7

Exclude the solutions that do not make $x - 3 - \sqrt{2 (15 - x)} = 0$ true.

$x = 7$

Solve by Factoring x-3 = square root of 30-2x

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