Solve for a square root of 41-8a+7=a

$\sqrt{41 - 8 a} + 7 = a$

Subtract $7$ from both sides of the equation.

$\sqrt{41 - 8 a} = a - 7$

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

${\sqrt{41 - 8 a}}^{2} = {(a - 7)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{41 - 8 a}$ as ${(41 - 8 a)}^{\frac{1}{2}}$ .

${({(41 - 8 a)}^{\frac{1}{2}})}^{2} = {(a - 7)}^{2}$

Simplify the left side.

Simplify ${({(41 - 8 a)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(41 - 8 a)}^{\frac{1}{2}})}^{2}$ .

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

${(41 - 8 a)}^{\frac{1}{2} \cdot 2} = {(a - 7)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(41 - 8 a)}^{\frac{1}{2} \cdot 2} = {(a - 7)}^{2}$

Rewrite the expression.

${(41 - 8 a)}^{1} = {(a - 7)}^{2}$

Simplify.

$41 - 8 a = {(a - 7)}^{2}$

Simplify the right side.

Simplify ${(a - 7)}^{2}$ .

Rewrite ${(a - 7)}^{2}$ as $(a - 7) (a - 7)$ .

$41 - 8 a = (a - 7) (a - 7)$

Expand $(a - 7) (a - 7)$ using the FOIL Method.

Apply the distributive property.

$41 - 8 a = a (a - 7) - 7 (a - 7)$

Apply the distributive property.

$41 - 8 a = a \cdot a + a \cdot - 7 - 7 (a - 7)$

Apply the distributive property.

$41 - 8 a = a \cdot a + a \cdot - 7 - 7 a - 7 \cdot - 7$

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

$41 - 8 a = a^{2} + a \cdot - 7 - 7 a - 7 \cdot - 7$

Move $- 7$ to the left of $a$ .

$41 - 8 a = a^{2} - 7 \cdot a - 7 a - 7 \cdot - 7$

Multiply $- 7$ by $- 7$ .

$41 - 8 a = a^{2} - 7 a - 7 a + 49$

Subtract $7 a$ from $- 7 a$ .

$41 - 8 a = a^{2} - 14 a + 49$

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.

$a^{2} - 14 a + 49 = 41 - 8 a$

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

Add $8 a$ to both sides of the equation.

$a^{2} - 14 a + 49 + 8 a = 41$

Add $- 14 a$ and $8 a$ .

$a^{2} - 6 a + 49 = 41$

Subtract $41$ from both sides of the equation.

$a^{2} - 6 a + 49 - 41 = 0$

Subtract $41$ from $49$ .

$a^{2} - 6 a + 8 = 0$

Factor $a^{2} - 6 a + 8$ 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Write the factored form using these integers.

$(a - 4) (a - 2) = 0$

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

$a - 4 = 0$

$a - 2 = 0$

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.

$a = 4$

Set $a - 2$ equal to $0$ and solve for $a$ .

Set $a - 2$ equal to $0$ .

$a - 2 = 0$

Add $2$ to both sides of the equation.

$a = 2$

The final solution is all the values that make $(a - 4) (a - 2) = 0$ true.

$a = 4, 2$

Exclude the solutions that do not make $\sqrt{41 - 8 a} + 7 = a$ true.

$No solution$

Solve for a square root of 41-8a+7=a

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