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

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

Step 1

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

Subtract $x$ from both sides of the equation.

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

Subtract $1$ from both sides of the equation.

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

Step 2

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

Add $x$ to both sides of the equation.

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

Add $1$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$x + 21 = (x + 1) (x + 1)$

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

Apply the distributive property.

$x + 21 = x (x + 1) + 1 (x + 1)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $x$ and $x$ .

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

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.

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $2 x$ .

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

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 $1$ .

$- 4, 5$

Write the factored form using these integers.

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

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

$x - 4 = 0$

$x + 5 = 0$

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 $x + 5$ equal to $0$ and solve for $x$ .

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

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

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

$x = 4, - 5$

Step 6

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

$x = 4$

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

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