Solve for x x-6 = square root of x-4

$x - 6 = \sqrt{x - 4}$

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.

$\sqrt{x - 4} = x - 6$

Step 2

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

${\sqrt{x - 4}}^{2} = {(x - 6)}^{2}$

Step 3

Simplify each side of the equation.

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

${({(x - 4)}^{\frac{1}{2}})}^{2} = {(x - 6)}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

${(x - 4)}^{\frac{1}{2} \cdot 2} = {(x - 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 4)}^{\frac{1}{2} \cdot 2} = {(x - 6)}^{2}$

Rewrite the expression.

${(x - 4)}^{1} = {(x - 6)}^{2}$

Simplify.

$x - 4 = {(x - 6)}^{2}$

Simplify the right side.

Simplify ${(x - 6)}^{2}$ .

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$x - 4 = (x - 6) (x - 6)$

Expand $(x - 6) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$x - 4 = x (x - 6) - 6 (x - 6)$

Apply the distributive property.

$x - 4 = x \cdot x + x \cdot - 6 - 6 (x - 6)$

Apply the distributive property.

$x - 4 = x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x - 4 = x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6$

Move $- 6$ to the left of $x$ .

$x - 4 = x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$x - 4 = x^{2} - 6 x - 6 x + 36$

Subtract $6 x$ from $- 6 x$ .

$x - 4 = x^{2} - 12 x + 36$

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.

$x^{2} - 12 x + 36 = x - 4$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 12 x + 36 - x = - 4$

Subtract $x$ from $- 12 x$ .

$x^{2} - 13 x + 36 = - 4$

Add $4$ to both sides of the equation.

$x^{2} - 13 x + 36 + 4 = 0$

Add $36$ and $4$ .

$x^{2} - 13 x + 40 = 0$

Factor $x^{2} - 13 x + 40$ 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 $40$ and whose sum is $- 13$ .

$- 8, - 5$

Write the factored form using these integers.

$(x - 8) (x - 5) = 0$

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

$x - 8 = 0$

$x - 5 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

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.

$x = 5$

The final solution is all the values that make $(x - 8) (x - 5) = 0$ true.

$x = 8, 5$

Step 5

Exclude the solutions that do not make $x - 6 = \sqrt{x - 4}$ true.

$x = 8$

Solve for x x-6 = square root of x-4

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