Solve by Factoring x-3 = square root of x-1

$x - 3 = \sqrt{x - 1}$

Step 1

Subtract $\sqrt{x - 1}$ from both sides of the equation.

$x - 3 - \sqrt{x - 1} = 0$

Step 2

Move all terms not containing $- \sqrt{x - 1}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- 3 - \sqrt{x - 1} = - x$

Add $3$ to both sides of the equation.

$- \sqrt{x - 1} = - x + 3$

Step 3

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

${(- \sqrt{x - 1})}^{2} = {(- x + 3)}^{2}$

Step 4

Simplify each side of the equation.

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

${(- {(x - 1)}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

Simplify the left side.

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

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

${(- 1)}^{2} {({(x - 1)}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

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

$1 {({(x - 1)}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

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

${({(x - 1)}^{\frac{1}{2}})}^{2} = {(- x + 3)}^{2}$

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

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

${(x - 1)}^{\frac{1}{2} \cdot 2} = {(- x + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 1)}^{\frac{1}{2} \cdot 2} = {(- x + 3)}^{2}$

Rewrite the expression.

${(x - 1)}^{1} = {(- x + 3)}^{2}$

Simplify.

$x - 1 = {(- x + 3)}^{2}$

Simplify the right side.

Simplify ${(- x + 3)}^{2}$ .

Rewrite ${(- x + 3)}^{2}$ as $(- x + 3) (- x + 3)$ .

$x - 1 = (- x + 3) (- x + 3)$

Expand $(- x + 3) (- x + 3)$ using the FOIL Method.

Apply the distributive property.

$x - 1 = - x (- x + 3) + 3 (- x + 3)$

Apply the distributive property.

$x - 1 = - x (- x) - x \cdot 3 + 3 (- x + 3)$

Apply the distributive property.

$x - 1 = - x (- x) - x \cdot 3 + 3 (- x) + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x - 1 = - 1 \cdot - 1 x \cdot x - x \cdot 3 + 3 (- x) + 3 \cdot 3$

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

Move $x$ .

$x - 1 = - 1 \cdot - 1 (x \cdot x) - x \cdot 3 + 3 (- x) + 3 \cdot 3$

Multiply $x$ by $x$ .

$x - 1 = - 1 \cdot - 1 x^{2} - x \cdot 3 + 3 (- x) + 3 \cdot 3$

Multiply $- 1$ by $- 1$ .

$x - 1 = 1 x^{2} - x \cdot 3 + 3 (- x) + 3 \cdot 3$

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

$x - 1 = x^{2} - x \cdot 3 + 3 (- x) + 3 \cdot 3$

Multiply $3$ by $- 1$ .

$x - 1 = x^{2} - 3 x + 3 (- x) + 3 \cdot 3$

Multiply $- 1$ by $3$ .

$x - 1 = x^{2} - 3 x - 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$x - 1 = x^{2} - 3 x - 3 x + 9$

Subtract $3 x$ from $- 3 x$ .

$x - 1 = x^{2} - 6 x + 9$

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.

$x^{2} - 6 x + 9 = x - 1$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 6 x + 9 - x = - 1$

Subtract $x$ from $- 6 x$ .

$x^{2} - 7 x + 9 = - 1$

Add $1$ to both sides of the equation.

$x^{2} - 7 x + 9 + 1 = 0$

Add $9$ and $1$ .

$x^{2} - 7 x + 10 = 0$

Factor $x^{2} - 7 x + 10$ 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$(x - 5) (x - 2) = 0$

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

$x - 5 = 0$

$x - 2 = 0$

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.

$x = 5$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x - 5) (x - 2) = 0$ true.

$x = 5, 2$

Step 6

Exclude the solutions that do not make $x - 3 - \sqrt{x - 1} = 0$ true.

$x = 5$

Solve by Factoring x-3 = square root of x-1

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