Solve for a a-6 = square root of a

$a - 6 = \sqrt{a}$

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.

$\sqrt{a} = a - 6$

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$a^{1} = {(a - 6)}^{2}$

Simplify.

$a = {(a - 6)}^{2}$

Simplify the right side.

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

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

$a = (a - 6) (a - 6)$

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

Apply the distributive property.

$a = a (a - 6) - 6 (a - 6)$

Apply the distributive property.

$a = a \cdot a + a \cdot - 6 - 6 (a - 6)$

Apply the distributive property.

$a = a \cdot a + a \cdot - 6 - 6 a - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

$a = a^{2} + a \cdot - 6 - 6 a - 6 \cdot - 6$

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

$a = a^{2} - 6 \cdot a - 6 a - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$a = a^{2} - 6 a - 6 a + 36$

Subtract $6 a$ from $- 6 a$ .

$a = a^{2} - 12 a + 36$

Step 4

Solve for $a$ .

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

$a^{2} - 12 a + 36 = a$

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

Subtract $a$ from both sides of the equation.

$a^{2} - 12 a + 36 - a = 0$

Subtract $a$ from $- 12 a$ .

$a^{2} - 13 a + 36 = 0$

Factor $a^{2} - 13 a + 36$ 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 $36$ and whose sum is $- 13$ .

$- 9, - 4$

Write the factored form using these integers.

$(a - 9) (a - 4) = 0$

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

$a - 9 = 0$

$a - 4 = 0$

Set $a - 9$ equal to $0$ and solve for $a$ .

Set $a - 9$ equal to $0$ .

$a - 9 = 0$

Add $9$ to both sides of the equation.

$a = 9$

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

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

$a - 4 = 0$

Add $4$ to both sides of the equation.

$a = 4$

The final solution is all the values that make $(a - 9) (a - 4) = 0$ true.

$a = 9, 4$

Step 5

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

$a = 9$

Solve for a a-6 = square root of a

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