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

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

Step 1

Subtract $14$ from both sides of the equation.

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

Step 2

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

Subtract $x$ from both sides of the equation.

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

Add $14$ to both sides of the equation.

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

Step 3

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 4

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

Apply the product rule to $7 x$ .

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $- 1$ .

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

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

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

Multiply $14$ by $- 1$ .

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

Multiply $- 1$ by $14$ .

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

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 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.

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

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 6

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

$x = 28$

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

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