Solve by Factoring square root of -14x+2=x-3

$\sqrt{- 14 x + 2} = x - 3$

Step 1

Move all the expressions to the left side of the equation.

Subtract $x$ from both sides of the equation.

$\sqrt{- 14 x + 2} - x = - 3$

Add $3$ to both sides of the equation.

$\sqrt{- 14 x + 2} - x + 3 = 0$

Step 2

Factor $2$ out of $- 14 x + 2$ .

Factor $2$ out of $- 14 x$ .

$\sqrt{2 (- 7 x) + 2} - x + 3 = 0$

Factor $2$ out of $2$ .

$\sqrt{2 (- 7 x) + 2 (1)} - x + 3 = 0$

Factor $2$ out of $2 (- 7 x) + 2 (1)$ .

$\sqrt{2 (- 7 x + 1)} - x + 3 = 0$

Step 3

Move all terms not containing $\sqrt{2 (- 7 x + 1)}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{2 (- 7 x + 1)} + 3 = x$

Subtract $3$ from both sides of the equation.

$\sqrt{2 (- 7 x + 1)} = x - 3$

Step 4

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

${\sqrt{2 (- 7 x + 1)}}^{2} = {(x - 3)}^{2}$

Step 5

Simplify each side of the equation.

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

${({(2 (- 7 x + 1))}^{\frac{1}{2}})}^{2} = {(x - 3)}^{2}$

Simplify the left side.

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

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

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

${(2 (- 7 x + 1))}^{\frac{1}{2} \cdot 2} = {(x - 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 (- 7 x + 1))}^{\frac{1}{2} \cdot 2} = {(x - 3)}^{2}$

Rewrite the expression.

${(2 (- 7 x + 1))}^{1} = {(x - 3)}^{2}$

Apply the distributive property.

${(2 (- 7 x) + 2 \cdot 1)}^{1} = {(x - 3)}^{2}$

Multiply.

Multiply $- 7$ by $2$ .

${(- 14 x + 2 \cdot 1)}^{1} = {(x - 3)}^{2}$

Simplify.

Multiply $2$ by $1$ .

${(- 14 x + 2)}^{1} = {(x - 3)}^{2}$

Simplify.

$- 14 x + 2 = {(x - 3)}^{2}$

Simplify the right side.

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

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

$- 14 x + 2 = (x - 3) (x - 3)$

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

Apply the distributive property.

$- 14 x + 2 = x (x - 3) - 3 (x - 3)$

Apply the distributive property.

$- 14 x + 2 = x \cdot x + x \cdot - 3 - 3 (x - 3)$

Apply the distributive property.

$- 14 x + 2 = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$- 14 x + 2 = x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3$

Move $- 3$ to the left of $x$ .

$- 14 x + 2 = x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3$

Multiply $- 3$ by $- 3$ .

$- 14 x + 2 = x^{2} - 3 x - 3 x + 9$

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

$- 14 x + 2 = x^{2} - 6 x + 9$

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.

$x^{2} - 6 x + 9 = - 14 x + 2$

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

Add $14 x$ to both sides of the equation.

$x^{2} - 6 x + 9 + 14 x = 2$

Add $- 6 x$ and $14 x$ .

$x^{2} + 8 x + 9 = 2$

Subtract $2$ from both sides of the equation.

$x^{2} + 8 x + 9 - 2 = 0$

Subtract $2$ from $9$ .

$x^{2} + 8 x + 7 = 0$

Factor $x^{2} + 8 x + 7$ 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 $7$ and whose sum is $8$ .

$1, 7$

Write the factored form using these integers.

$(x + 1) (x + 7) = 0$

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

$x + 1 = 0$

$x + 7 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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.

$x = - 7$

The final solution is all the values that make $(x + 1) (x + 7) = 0$ true.

$x = - 1, - 7$

Step 7

Exclude the solutions that do not make $\sqrt{2 (- 7 x + 1)} - x + 3 = 0$ true.

$No solution$

Solve by Factoring square root of -14x+2=x-3

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