Algebra Examples

$x + \sqrt{x} = 20$

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.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

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

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $20$ by $20$ .

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

Step 3.3.1.3.1.2

Multiply $- 1$ by $20$ .

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

Step 3.3.1.3.1.3

Multiply $20$ by $- 1$ .

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

Step 3.3.1.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.3.1.3.1.5

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

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Step 3.3.1.3.1.5.1

Move $x$ .

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

Step 3.3.1.3.1.5.2

Multiply $x$ by $x$ .

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

Step 3.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.3.1.7

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

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

Step 3.3.1.3.2

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

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

Step 4

Solve for $x$ .

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Step 4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 4.2

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

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Step 4.2.1

Subtract $x$ from both sides of the equation.

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

Step 4.2.2

Subtract $x$ from $- 40 x$ .

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

Step 4.3

Factor $400 + x^{2} - 41 x$ using the AC method.

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Step 4.3.1

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 $- 41$ .

$- 25, - 16$

Step 4.3.2

Write the factored form using these integers.

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

Step 4.4

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

$x - 25 = 0$

$x - 16 = 0$

Step 4.5

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

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Step 4.5.1

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

$x - 25 = 0$

Step 4.5.2

Add $25$ to both sides of the equation.

$x = 25$

Step 4.6

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

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Step 4.6.1

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

$x - 16 = 0$

Step 4.6.2

Add $16$ to both sides of the equation.

$x = 16$

Step 4.7

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.

$x = 16$