Solve by Factoring 4+ square root of 5x+66=x+10

$4 + \sqrt{5 x + 66} = x + 10$

Step 1

Move all the expressions to the left side of the equation.

Subtract $x$ from both sides of the equation.

$4 + \sqrt{5 x + 66} - x = 10$

Subtract $10$ from both sides of the equation.

$4 + \sqrt{5 x + 66} - x - 10 = 0$

Step 2

Subtract $10$ from $4$ .

$- 6 + \sqrt{5 x + 66} - x = 0$

Step 3

Move all terms not containing $\sqrt{5 x + 66}$ to the right side of the equation.

Add $6$ to both sides of the equation.

$\sqrt{5 x + 66} - x = 6$

Add $x$ to both sides of the equation.

$\sqrt{5 x + 66} = 6 + x$

Step 4

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

${\sqrt{5 x + 66}}^{2} = {(6 + x)}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 66}$ as ${(5 x + 66)}^{\frac{1}{2}}$ .

${({(5 x + 66)}^{\frac{1}{2}})}^{2} = {(6 + x)}^{2}$

Simplify the left side.

Simplify ${({(5 x + 66)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 x + 66)}^{\frac{1}{2}})}^{2}$ .

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

${(5 x + 66)}^{\frac{1}{2} \cdot 2} = {(6 + x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(5 x + 66)}^{\frac{1}{2} \cdot 2} = {(6 + x)}^{2}$

Rewrite the expression.

${(5 x + 66)}^{1} = {(6 + x)}^{2}$

Simplify.

$5 x + 66 = {(6 + x)}^{2}$

Simplify the right side.

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

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

$5 x + 66 = (6 + x) (6 + x)$

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

Apply the distributive property.

$5 x + 66 = 6 (6 + x) + x (6 + x)$

Apply the distributive property.

$5 x + 66 = 6 \cdot 6 + 6 x + x (6 + x)$

Apply the distributive property.

$5 x + 66 = 6 \cdot 6 + 6 x + x \cdot 6 + x \cdot x$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$5 x + 66 = 36 + 6 x + x \cdot 6 + x \cdot x$

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

$5 x + 66 = 36 + 6 x + 6 \cdot x + x \cdot x$

Multiply $x$ by $x$ .

$5 x + 66 = 36 + 6 x + 6 x + x^{2}$

Add $6 x$ and $6 x$ .

$5 x + 66 = 36 + 12 x + x^{2}$

Step 6

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.

$36 + 12 x + x^{2} = 5 x + 66$

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

Subtract $5 x$ from both sides of the equation.

$36 + 12 x + x^{2} - 5 x = 66$

Subtract $5 x$ from $12 x$ .

$36 + x^{2} + 7 x = 66$

Subtract $66$ from both sides of the equation.

$36 + x^{2} + 7 x - 66 = 0$

Subtract $66$ from $36$ .

$x^{2} + 7 x - 30 = 0$

Factor $x^{2} + 7 x - 30$ 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 $- 30$ and whose sum is $7$ .

$- 3, 10$

Write the factored form using these integers.

$(x - 3) (x + 10) = 0$

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

$x - 3 = 0$

$x + 10 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 10$ equal to $0$ and solve for $x$ .

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Subtract $10$ from both sides of the equation.

$x = - 10$

The final solution is all the values that make $(x - 3) (x + 10) = 0$ true.

$x = 3, - 10$

Step 7

Exclude the solutions that do not make $- 6 + \sqrt{5 x + 66} - x = 0$ true.

$x = 3$

Solve by Factoring 4+ square root of 5x+66=x+10

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