Solve by Factoring square root of 53-x=x+3

$\sqrt{53 - x} = x + 3$

Step 1

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

Subtract $x$ from both sides of the equation.

$\sqrt{53 - x} - x = 3$

Subtract $3$ from both sides of the equation.

$\sqrt{53 - x} - x - 3 = 0$

Step 2

Move all terms not containing $\sqrt{53 - x}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{53 - x} - 3 = x$

Add $3$ to both sides of the equation.

$\sqrt{53 - x} = x + 3$

Step 3

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

${\sqrt{53 - x}}^{2} = {(x + 3)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{53 - x}$ as ${(53 - x)}^{\frac{1}{2}}$ .

${({(53 - x)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Simplify the left side.

Simplify ${({(53 - x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(53 - x)}^{\frac{1}{2}})}^{2}$ .

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

${(53 - x)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(53 - x)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Rewrite the expression.

${(53 - x)}^{1} = {(x + 3)}^{2}$

Simplify.

$53 - x = {(x + 3)}^{2}$

Simplify the right side.

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

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

$53 - x = (x + 3) (x + 3)$

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

Apply the distributive property.

$53 - x = x (x + 3) + 3 (x + 3)$

Apply the distributive property.

$53 - x = x \cdot x + x \cdot 3 + 3 (x + 3)$

Apply the distributive property.

$53 - x = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$53 - x = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Move $3$ to the left of $x$ .

$53 - x = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$53 - x = x^{2} + 3 x + 3 x + 9$

Add $3 x$ and $3 x$ .

$53 - x = x^{2} + 6 x + 9$

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.

$x^{2} + 6 x + 9 = 53 - x$

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

Add $x$ to both sides of the equation.

$x^{2} + 6 x + 9 + x = 53$

Add $6 x$ and $x$ .

$x^{2} + 7 x + 9 = 53$

Subtract $53$ from both sides of the equation.

$x^{2} + 7 x + 9 - 53 = 0$

Subtract $53$ from $9$ .

$x^{2} + 7 x - 44 = 0$

Factor $x^{2} + 7 x - 44$ 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 $- 44$ and whose sum is $7$ .

$- 4, 11$

Write the factored form using these integers.

$(x - 4) (x + 11) = 0$

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

$x - 4 = 0$

$x + 11 = 0$

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 $x + 11$ equal to $0$ and solve for $x$ .

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Subtract $11$ from both sides of the equation.

$x = - 11$

The final solution is all the values that make $(x - 4) (x + 11) = 0$ true.

$x = 4, - 11$

Step 6

Exclude the solutions that do not make $\sqrt{53 - x} - x - 3 = 0$ true.

$x = 4$

Solve by Factoring square root of 53-x=x+3

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