Solve for x 4-x = square root of x-4

$4 - x = \sqrt{x - 4}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{x - 4} = 4 - x$

Step 2

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

${\sqrt{x - 4}}^{2} = {(4 - x)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

${({(x - 4)}^{\frac{1}{2}})}^{2} = {(4 - x)}^{2}$

Simplify the left side.

Simplify ${({(x - 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x - 4)}^{\frac{1}{2}})}^{2}$ .

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

${(x - 4)}^{\frac{1}{2} \cdot 2} = {(4 - x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 4)}^{\frac{1}{2} \cdot 2} = {(4 - x)}^{2}$

Rewrite the expression.

${(x - 4)}^{1} = {(4 - x)}^{2}$

Simplify.

$x - 4 = {(4 - x)}^{2}$

Simplify the right side.

Simplify ${(4 - x)}^{2}$ .

Rewrite ${(4 - x)}^{2}$ as $(4 - x) (4 - x)$ .

$x - 4 = (4 - x) (4 - x)$

Expand $(4 - x) (4 - x)$ using the FOIL Method.

Apply the distributive property.

$x - 4 = 4 (4 - x) - x (4 - x)$

Apply the distributive property.

$x - 4 = 4 \cdot 4 + 4 (- x) - x (4 - x)$

Apply the distributive property.

$x - 4 = 4 \cdot 4 + 4 (- x) - x \cdot 4 - x (- x)$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$x - 4 = 16 + 4 (- x) - x \cdot 4 - x (- x)$

Multiply $- 1$ by $4$ .

$x - 4 = 16 - 4 x - x \cdot 4 - x (- x)$

Multiply $4$ by $- 1$ .

$x - 4 = 16 - 4 x - 4 x - x (- x)$

Rewrite using the commutative property of multiplication.

$x - 4 = 16 - 4 x - 4 x - 1 \cdot - 1 x \cdot x$

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

Move $x$ .

$x - 4 = 16 - 4 x - 4 x - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$x - 4 = 16 - 4 x - 4 x - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$x - 4 = 16 - 4 x - 4 x + 1 x^{2}$

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

$x - 4 = 16 - 4 x - 4 x + x^{2}$

Subtract $4 x$ from $- 4 x$ .

$x - 4 = 16 - 8 x + x^{2}$

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.

$16 - 8 x + x^{2} = x - 4$

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

Subtract $x$ from both sides of the equation.

$16 - 8 x + x^{2} - x = - 4$

Subtract $x$ from $- 8 x$ .

$16 + x^{2} - 9 x = - 4$

Add $4$ to both sides of the equation.

$16 + x^{2} - 9 x + 4 = 0$

Add $16$ and $4$ .

$x^{2} - 9 x + 20 = 0$

Factor $x^{2} - 9 x + 20$ 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 $20$ and whose sum is $- 9$ .

$- 5, - 4$

Write the factored form using these integers.

$(x - 5) (x - 4) = 0$

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

$x - 5 = 0$

$x - 4 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(x - 5) (x - 4) = 0$ true.

$x = 5, 4$

Step 5

Exclude the solutions that do not make $4 - x = \sqrt{x - 4}$ true.

$x = 4$

Solve for x 4-x = square root of x-4

Related Questions

Support LCMGCF.com by disabling your adblocker.