Solve for x 9=x+2 square root of x-1

$9 = x + 2 \sqrt{x - 1}$

Rewrite the equation as $x + 2 \sqrt{x - 1} = 9$ .

$x + 2 \sqrt{x - 1} = 9$

Subtract $x$ from both sides of the equation.

$2 \sqrt{x - 1} = 9 - x$

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

${(2 \sqrt{x - 1})}^{2} = {(9 - x)}^{2}$

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

$4 {({(x - 1)}^{\frac{1}{2}})}^{2} = {(9 - x)}^{2}$

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

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

$4 {(x - 1)}^{\frac{1}{2} \cdot 2} = {(9 - x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(x - 1)}^{\frac{1}{2} \cdot 2} = {(9 - x)}^{2}$

Rewrite the expression.

$4 {(x - 1)}^{1} = {(9 - x)}^{2}$

Simplify.

$4 (x - 1) = {(9 - x)}^{2}$

Apply the distributive property.

$4 x + 4 \cdot - 1 = {(9 - x)}^{2}$

Multiply $4$ by $- 1$ .

$4 x - 4 = {(9 - x)}^{2}$

Simplify the right side.

Simplify ${(9 - x)}^{2}$ .

Rewrite ${(9 - x)}^{2}$ as $(9 - x) (9 - x)$ .

$4 x - 4 = (9 - x) (9 - x)$

Expand $(9 - x) (9 - x)$ using the FOIL Method.

Apply the distributive property.

$4 x - 4 = 9 (9 - x) - x (9 - x)$

Apply the distributive property.

$4 x - 4 = 9 \cdot 9 + 9 (- x) - x (9 - x)$

Apply the distributive property.

$4 x - 4 = 9 \cdot 9 + 9 (- x) - x \cdot 9 - x (- x)$

Simplify and combine like terms.

Simplify each term.

Multiply $9$ by $9$ .

$4 x - 4 = 81 + 9 (- x) - x \cdot 9 - x (- x)$

Multiply $- 1$ by $9$ .

$4 x - 4 = 81 - 9 x - x \cdot 9 - x (- x)$

Multiply $9$ by $- 1$ .

$4 x - 4 = 81 - 9 x - 9 x - x (- x)$

Rewrite using the commutative property of multiplication.

$4 x - 4 = 81 - 9 x - 9 x - 1 \cdot - 1 x \cdot x$

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

Move $x$ .

$4 x - 4 = 81 - 9 x - 9 x - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$4 x - 4 = 81 - 9 x - 9 x - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$4 x - 4 = 81 - 9 x - 9 x + 1 x^{2}$

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

$4 x - 4 = 81 - 9 x - 9 x + x^{2}$

Subtract $9 x$ from $- 9 x$ .

$4 x - 4 = 81 - 18 x + x^{2}$

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.

$81 - 18 x + x^{2} = 4 x - 4$

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

Subtract $4 x$ from both sides of the equation.

$81 - 18 x + x^{2} - 4 x = - 4$

Subtract $4 x$ from $- 18 x$ .

$81 + x^{2} - 22 x = - 4$

Add $4$ to both sides of the equation.

$81 + x^{2} - 22 x + 4 = 0$

Add $81$ and $4$ .

$x^{2} - 22 x + 85 = 0$

Factor $x^{2} - 22 x + 85$ 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 $85$ and whose sum is $- 22$ .

$- 17, - 5$

Write the factored form using these integers.

$(x - 17) (x - 5) = 0$

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

$x - 17 = 0$

$x - 5 = 0$

Set $x - 17$ equal to $0$ and solve for $x$ .

Set $x - 17$ equal to $0$ .

$x - 17 = 0$

Add $17$ to both sides of the equation.

$x = 17$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $(x - 17) (x - 5) = 0$ true.

$x = 17, 5$

Exclude the solutions that do not make $9 = x + 2 \sqrt{x - 1}$ true.

$x = 5$

Solve for x 9=x+2 square root of x-1

Related Questions

Support LCMGCF.com by disabling your adblocker.