Solve by Factoring square root of 7x-x+14=0

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

Step 1

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

Add $x$ to both sides of the equation.

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

Subtract $14$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$7 x = x \cdot x + x \cdot - 14 - 14 x - 14 \cdot - 14$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 14$ to the left of $x$ .

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

Multiply $- 14$ by $- 14$ .

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

Subtract $14 x$ from $- 14 x$ .

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

Step 4

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.

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

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

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

$x^{2} - 28 x + 196 - 7 x = 0$

Subtract $7 x$ from $- 28 x$ .

$x^{2} - 35 x + 196 = 0$

Factor $x^{2} - 35 x + 196$ 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 $196$ and whose sum is $- 35$ .

$- 28, - 7$

Write the factored form using these integers.

$(x - 28) (x - 7) = 0$

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

$x - 28 = 0$

$x - 7 = 0$

Set $x - 28$ equal to $0$ and solve for $x$ .

Set $x - 28$ equal to $0$ .

$x - 28 = 0$

Add $28$ to both sides of the equation.

$x = 28$

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.

$x = 7$

The final solution is all the values that make $(x - 28) (x - 7) = 0$ true.

$x = 28, 7$

Step 5

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

$x = 28$

Solve by Factoring square root of 7x-x+14=0

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