Solve by Factoring x = square root of 4x+21

$x = \sqrt{4 x + 21}$

Step 1

Subtract $\sqrt{4 x + 21}$ from both sides of the equation.

$x - \sqrt{4 x + 21} = 0$

Step 2

Subtract $x$ from both sides of the equation.

$- \sqrt{4 x + 21} = - x$

Step 3

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

${(- \sqrt{4 x + 21})}^{2} = {(- x)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x + 21}$ as ${(4 x + 21)}^{\frac{1}{2}}$ .

${(- {(4 x + 21)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Simplify the left side.

Simplify ${(- {(4 x + 21)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(4 x + 21)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(4 x + 21)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

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

$1 {({(4 x + 21)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Multiply ${({(4 x + 21)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(4 x + 21)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Multiply the exponents in ${({(4 x + 21)}^{\frac{1}{2}})}^{2}$ .

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

${(4 x + 21)}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x + 21)}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Rewrite the expression.

${(4 x + 21)}^{1} = {(- x)}^{2}$

Simplify.

$4 x + 21 = {(- x)}^{2}$

Simplify the right side.

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

Apply the product rule to $- x$ .

$4 x + 21 = {(- 1)}^{2} x^{2}$

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

$4 x + 21 = 1 x^{2}$

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

$4 x + 21 = x^{2}$

Step 5

Solve for $x$ .

Subtract $x^{2}$ from both sides of the equation.

$4 x + 21 - x^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $4 x + 21 - x^{2}$ .

Reorder the expression.

Move $21$ .

$4 x - x^{2} + 21 = 0$

Reorder $4 x$ and $- x^{2}$ .

$- x^{2} + 4 x + 21 = 0$

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 4 x + 21 = 0$

Factor $- 1$ out of $4 x$ .

$- (x^{2}) - (- 4 x) + 21 = 0$

Rewrite $21$ as $- 1 (- 21)$ .

$- (x^{2}) - (- 4 x) - 1 \cdot - 21 = 0$

Factor $- 1$ out of $- (x^{2}) - (- 4 x)$ .

$- (x^{2} - 4 x) - 1 \cdot - 21 = 0$

Factor $- 1$ out of $- (x^{2} - 4 x) - 1 (- 21)$ .

$- (x^{2} - 4 x - 21) = 0$

Factor.

Factor $x^{2} - 4 x - 21$ 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 $- 21$ and whose sum is $- 4$ .

$- 7, 3$

Write the factored form using these integers.

$- ((x - 7) (x + 3)) = 0$

Remove unnecessary parentheses.

$- (x - 7) (x + 3) = 0$

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

$x - 7 = 0$

$x + 3 = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

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 $- (x - 7) (x + 3) = 0$ true.

$x = 7, - 3$

Step 6

Exclude the solutions that do not make $x - \sqrt{4 x + 21} = 0$ true.

$x = 7$

Solve by Factoring x = square root of 4x+21

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