Solve for x x+ square root of x=20

$x + \sqrt{x} = 20$

Step 1

Subtract $x$ from both sides of the equation.

$\sqrt{x} = 20 - x$

Step 2

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

${\sqrt{x}}^{2} = {(20 - x)}^{2}$

Step 3

Simplify each side of the equation.

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

${(x^{\frac{1}{2}})}^{2} = {(20 - x)}^{2}$

Simplify the left side.

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

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

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

$x^{\frac{1}{2} \cdot 2} = {(20 - x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(20 - x)}^{2}$

Rewrite the expression.

$x^{1} = {(20 - x)}^{2}$

Simplify.

$x = {(20 - x)}^{2}$

Simplify the right side.

Simplify ${(20 - x)}^{2}$ .

Rewrite ${(20 - x)}^{2}$ as $(20 - x) (20 - x)$ .

$x = (20 - x) (20 - x)$

Expand $(20 - x) (20 - x)$ using the FOIL Method.

Apply the distributive property.

$x = 20 (20 - x) - x (20 - x)$

Apply the distributive property.

$x = 20 \cdot 20 + 20 (- x) - x (20 - x)$

Apply the distributive property.

$x = 20 \cdot 20 + 20 (- x) - x \cdot 20 - x (- x)$

Simplify and combine like terms.

Simplify each term.

Multiply $20$ by $20$ .

$x = 400 + 20 (- x) - x \cdot 20 - x (- x)$

Multiply $- 1$ by $20$ .

$x = 400 - 20 x - x \cdot 20 - x (- x)$

Multiply $20$ by $- 1$ .

$x = 400 - 20 x - 20 x - x (- x)$

Rewrite using the commutative property of multiplication.

$x = 400 - 20 x - 20 x - 1 \cdot - 1 x \cdot x$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x = 400 - 20 x - 20 x - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$x = 400 - 20 x - 20 x - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$x = 400 - 20 x - 20 x + 1 x^{2}$

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

$x = 400 - 20 x - 20 x + x^{2}$

Subtract $20 x$ from $- 20 x$ .

$x = 400 - 40 x + x^{2}$

Step 4

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.

$400 - 40 x + x^{2} = x$

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

Subtract $x$ from both sides of the equation.

$400 - 40 x + x^{2} - x = 0$

Subtract $x$ from $- 40 x$ .

$400 + x^{2} - 41 x = 0$

Factor $400 + x^{2} - 41 x$ 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 $400$ and whose sum is $- 41$ .

$- 25, - 16$

Write the factored form using these integers.

$(x - 25) (x - 16) = 0$

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

$x - 25 = 0$

$x - 16 = 0$

Set $x - 25$ equal to $0$ and solve for $x$ .

Set $x - 25$ equal to $0$ .

$x - 25 = 0$

Add $25$ to both sides of the equation.

$x = 25$

Set $x - 16$ equal to $0$ and solve for $x$ .

Set $x - 16$ equal to $0$ .

$x - 16 = 0$

Add $16$ to both sides of the equation.

$x = 16$

The final solution is all the values that make $(x - 25) (x - 16) = 0$ true.

$x = 25, 16$

Step 5

Exclude the solutions that do not make $x + \sqrt{x} = 20$ true.

$x = 16$

Solve for x x+ square root of x=20

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