Solve for x square root of 13-x=x-1

$\sqrt{13 - x} = x - 1$

Step 1

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

${\sqrt{13 - x}}^{2} = {(x - 1)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13 - x}$ as ${(13 - x)}^{\frac{1}{2}}$ .

${({(13 - x)}^{\frac{1}{2}})}^{2} = {(x - 1)}^{2}$

Simplify the left side.

Simplify ${({(13 - x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(13 - x)}^{\frac{1}{2}})}^{2}$ .

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

${(13 - x)}^{\frac{1}{2} \cdot 2} = {(x - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(13 - x)}^{\frac{1}{2} \cdot 2} = {(x - 1)}^{2}$

Rewrite the expression.

${(13 - x)}^{1} = {(x - 1)}^{2}$

Simplify.

$13 - x = {(x - 1)}^{2}$

Simplify the right side.

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

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$13 - x = (x - 1) (x - 1)$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$13 - x = x (x - 1) - 1 (x - 1)$

Apply the distributive property.

$13 - x = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Apply the distributive property.

$13 - x = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$13 - x = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1$

Move $- 1$ to the left of $x$ .

$13 - x = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1$

Rewrite $- 1 x$ as $- x$ .

$13 - x = x^{2} - x - 1 x - 1 \cdot - 1$

Rewrite $- 1 x$ as $- x$ .

$13 - x = x^{2} - x - x - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$13 - x = x^{2} - x - x + 1$

Subtract $x$ from $- x$ .

$13 - x = x^{2} - 2 x + 1$

Step 3

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.

$x^{2} - 2 x + 1 = 13 - x$

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

Add $x$ to both sides of the equation.

$x^{2} - 2 x + 1 + x = 13$

Add $- 2 x$ and $x$ .

$x^{2} - x + 1 = 13$

Subtract $13$ from both sides of the equation.

$x^{2} - x + 1 - 13 = 0$

Subtract $13$ from $1$ .

$x^{2} - x - 12 = 0$

Factor $x^{2} - x - 12$ 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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Write the factored form using these integers.

$(x - 4) (x + 3) = 0$

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

$x - 4 = 0$

$x + 3 = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $(x - 4) (x + 3) = 0$ true.

$x = 4, - 3$

Step 4

Exclude the solutions that do not make $\sqrt{13 - x} = x - 1$ true.

$x = 4$

Solve for x square root of 13-x=x-1

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