Solve for a square root of -36+13a=a

$\sqrt{- 36 + 13 a} = a$

Step 1

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

${\sqrt{- 36 + 13 a}}^{2} = a^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 36 + 13 a}$ as ${(- 36 + 13 a)}^{\frac{1}{2}}$ .

${({(- 36 + 13 a)}^{\frac{1}{2}})}^{2} = a^{2}$

Simplify the left side.

Simplify ${({(- 36 + 13 a)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(- 36 + 13 a)}^{\frac{1}{2}})}^{2}$ .

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

${(- 36 + 13 a)}^{\frac{1}{2} \cdot 2} = a^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- 36 + 13 a)}^{\frac{1}{2} \cdot 2} = a^{2}$

Rewrite the expression.

${(- 36 + 13 a)}^{1} = a^{2}$

Simplify.

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

Step 3

Solve for $a$ .

Subtract $a^{2}$ from both sides of the equation.

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

Factor the left side of the equation.

Factor $- 1$ out of $- 36 + 13 a - a^{2}$ .

Reorder the expression.

Move $- 36$ .

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

Reorder $13 a$ and $- a^{2}$ .

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

Factor $- 1$ out of $- a^{2}$ .

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

Factor $- 1$ out of $13 a$ .

$- (a^{2}) - (- 13 a) - 36 = 0$

Rewrite $- 36$ as $- 1 (36)$ .

$- (a^{2}) - (- 13 a) - 1 \cdot 36 = 0$

Factor $- 1$ out of $- (a^{2}) - (- 13 a)$ .

$- (a^{2} - 13 a) - 1 \cdot 36 = 0$

Factor $- 1$ out of $- (a^{2} - 13 a) - 1 (36)$ .

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

Factor.

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.

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

Remove unnecessary parentheses.

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

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.

$a = 9, 4$

Solve for a square root of -36+13a=a

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