Solve by Factoring square root of 21-x=x-1

$\sqrt{21 - x} = x - 1$

Step 1

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

Subtract $x$ from both sides of the equation.

$\sqrt{21 - x} - x = - 1$

Add $1$ to both sides of the equation.

$\sqrt{21 - x} - x + 1 = 0$

Step 2

Move all terms not containing $\sqrt{21 - x}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{21 - x} + 1 = x$

Subtract $1$ from both sides of the equation.

$\sqrt{21 - x} = x - 1$

Step 3

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

${\sqrt{21 - x}}^{2} = {(x - 1)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21 - x}$ as ${(21 - x)}^{\frac{1}{2}}$ .

${({(21 - x)}^{\frac{1}{2}})}^{2} = {(x - 1)}^{2}$

Simplify the left side.

Simplify ${({(21 - x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(21 - x)}^{\frac{1}{2}})}^{2}$ .

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

${(21 - x)}^{\frac{1}{2} \cdot 2} = {(x - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(21 - x)}^{\frac{1}{2} \cdot 2} = {(x - 1)}^{2}$

Rewrite the expression.

${(21 - x)}^{1} = {(x - 1)}^{2}$

Simplify.

$21 - x = {(x - 1)}^{2}$

Simplify the right side.

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

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

$21 - x = (x - 1) (x - 1)$

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

Apply the distributive property.

$21 - x = x (x - 1) - 1 (x - 1)$

Apply the distributive property.

$21 - x = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Apply the distributive property.

$21 - x = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$21 - x = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1$

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

$21 - x = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1$

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

$21 - x = x^{2} - x - 1 x - 1 \cdot - 1$

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

$21 - x = x^{2} - x - x - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$21 - x = x^{2} - x - x + 1$

Subtract $x$ from $- x$ .

$21 - x = x^{2} - 2 x + 1$

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.

$x^{2} - 2 x + 1 = 21 - x$

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

Add $x$ to both sides of the equation.

$x^{2} - 2 x + 1 + x = 21$

Add $- 2 x$ and $x$ .

$x^{2} - x + 1 = 21$

Subtract $21$ from both sides of the equation.

$x^{2} - x + 1 - 21 = 0$

Subtract $21$ from $1$ .

$x^{2} - x - 20 = 0$

Factor $x^{2} - x - 20$ 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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Write the factored form using these integers.

$(x - 5) (x + 4) = 0$

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

$x - 5 = 0$

$x + 4 = 0$

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 $x + 4$ equal to $0$ and solve for $x$ .

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(x - 5) (x + 4) = 0$ true.

$x = 5, - 4$

Step 6

Exclude the solutions that do not make $\sqrt{21 - x} - x + 1 = 0$ true.

$x = 5$

Solve by Factoring square root of 21-x=x-1

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