Solve by Factoring x- square root of 10-9x=-20

$x - \sqrt{10 - 9 x} = - 20$

Step 1

Add $20$ to both sides of the equation.

$x - \sqrt{10 - 9 x} + 20 = 0$

Step 2

Move all terms not containing $- \sqrt{10 - 9 x}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- \sqrt{10 - 9 x} + 20 = - x$

Subtract $20$ from both sides of the equation.

$- \sqrt{10 - 9 x} = - x - 20$

Step 3

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

${(- \sqrt{10 - 9 x})}^{2} = {(- x - 20)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 - 9 x}$ as ${(10 - 9 x)}^{\frac{1}{2}}$ .

${(- {(10 - 9 x)}^{\frac{1}{2}})}^{2} = {(- x - 20)}^{2}$

Simplify the left side.

Simplify ${(- {(10 - 9 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(10 - 9 x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(10 - 9 x)}^{\frac{1}{2}})}^{2} = {(- x - 20)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(10 - 9 x)}^{\frac{1}{2}})}^{2} = {(- x - 20)}^{2}$

Multiply ${({(10 - 9 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(10 - 9 x)}^{\frac{1}{2}})}^{2} = {(- x - 20)}^{2}$

Multiply the exponents in ${({(10 - 9 x)}^{\frac{1}{2}})}^{2}$ .

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

${(10 - 9 x)}^{\frac{1}{2} \cdot 2} = {(- x - 20)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(10 - 9 x)}^{\frac{1}{2} \cdot 2} = {(- x - 20)}^{2}$

Rewrite the expression.

${(10 - 9 x)}^{1} = {(- x - 20)}^{2}$

Simplify.

$10 - 9 x = {(- x - 20)}^{2}$

Simplify the right side.

Simplify ${(- x - 20)}^{2}$ .

Rewrite ${(- x - 20)}^{2}$ as $(- x - 20) (- x - 20)$ .

$10 - 9 x = (- x - 20) (- x - 20)$

Expand $(- x - 20) (- x - 20)$ using the FOIL Method.

Apply the distributive property.

$10 - 9 x = - x (- x - 20) - 20 (- x - 20)$

Apply the distributive property.

$10 - 9 x = - x (- x) - x \cdot - 20 - 20 (- x - 20)$

Apply the distributive property.

$10 - 9 x = - x (- x) - x \cdot - 20 - 20 (- x) - 20 \cdot - 20$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$10 - 9 x = - 1 \cdot - 1 x \cdot x - x \cdot - 20 - 20 (- x) - 20 \cdot - 20$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$10 - 9 x = - 1 \cdot - 1 (x \cdot x) - x \cdot - 20 - 20 (- x) - 20 \cdot - 20$

Multiply $x$ by $x$ .

$10 - 9 x = - 1 \cdot - 1 x^{2} - x \cdot - 20 - 20 (- x) - 20 \cdot - 20$

Multiply $- 1$ by $- 1$ .

$10 - 9 x = 1 x^{2} - x \cdot - 20 - 20 (- x) - 20 \cdot - 20$

Multiply $x^{2}$ by $1$ .

$10 - 9 x = x^{2} - x \cdot - 20 - 20 (- x) - 20 \cdot - 20$

Multiply $- 20$ by $- 1$ .

$10 - 9 x = x^{2} + 20 x - 20 (- x) - 20 \cdot - 20$

Multiply $- 1$ by $- 20$ .

$10 - 9 x = x^{2} + 20 x + 20 x - 20 \cdot - 20$

Multiply $- 20$ by $- 20$ .

$10 - 9 x = x^{2} + 20 x + 20 x + 400$

Add $20 x$ and $20 x$ .

$10 - 9 x = x^{2} + 40 x + 400$

Step 5

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.

$x^{2} + 40 x + 400 = 10 - 9 x$

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

Add $9 x$ to both sides of the equation.

$x^{2} + 40 x + 400 + 9 x = 10$

Add $40 x$ and $9 x$ .

$x^{2} + 49 x + 400 = 10$

Subtract $10$ from both sides of the equation.

$x^{2} + 49 x + 400 - 10 = 0$

Subtract $10$ from $400$ .

$x^{2} + 49 x + 390 = 0$

Factor $x^{2} + 49 x + 390$ 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 $390$ and whose sum is $49$ .

$10, 39$

Write the factored form using these integers.

$(x + 10) (x + 39) = 0$

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

$x + 10 = 0$

$x + 39 = 0$

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.

$x = - 10$

Set $x + 39$ equal to $0$ and solve for $x$ .

Set $x + 39$ equal to $0$ .

$x + 39 = 0$

Subtract $39$ from both sides of the equation.

$x = - 39$

The final solution is all the values that make $(x + 10) (x + 39) = 0$ true.

$x = - 10, - 39$

Step 6

Exclude the solutions that do not make $x - \sqrt{10 - 9 x} + 20 = 0$ true.

$x = - 10$

Solve by Factoring x- square root of 10-9x=-20

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