Algebra Examples

$x (18 - x) = 2 (9 x - 32)$

Step 1

Simplify $x (18 - x)$ .

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Step 1.1

Rewrite.

$0 + 0 + x (18 - x) = 2 (9 x - 32)$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$x \cdot 18 + x (- x) = 2 (9 x - 32)$

Step 1.2.2

Reorder.

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Step 1.2.2.1

Move $18$ to the left of $x$ .

$18 \cdot x + x (- x) = 2 (9 x - 32)$

Step 1.2.2.2

Rewrite using the commutative property of multiplication.

$18 \cdot x - x \cdot x = 2 (9 x - 32)$

Step 1.3

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

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Step 1.3.1

Move $x$ .

$18 x - (x \cdot x) = 2 (9 x - 32)$

Step 1.3.2

Multiply $x$ by $x$ .

$18 x - x^{2} = 2 (9 x - 32)$

Step 2

Simplify $2 (9 x - 32)$ .

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Step 2.1

Apply the distributive property.

$18 x - x^{2} = 2 (9 x) + 2 \cdot - 32$

Step 2.2

Multiply.

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Step 2.2.1

Multiply $9$ by $2$ .

$18 x - x^{2} = 18 x + 2 \cdot - 32$

Step 2.2.2

Multiply $2$ by $- 32$ .

$18 x - x^{2} = 18 x - 64$

Step 3

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

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Step 3.1

Subtract $18 x$ from both sides of the equation.

$18 x - x^{2} - 18 x = - 64$

Step 3.2

Combine the opposite terms in $18 x - x^{2} - 18 x$ .

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Step 3.2.1

Subtract $18 x$ from $18 x$ .

$- x^{2} + 0 = - 64$

Step 3.2.2

Add $- x^{2}$ and $0$ .

$- x^{2} = - 64$

Step 4

Divide each term in $- x^{2} = - 64$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- x^{2} = - 64$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 64}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 64}{- 1}$

Step 4.2.2

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

$x^{2} = \frac{- 64}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 64$ by $- 1$ .

$x^{2} = 64$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{64}$

Step 6

Simplify $\pm \sqrt{64}$ .

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Step 6.1

Rewrite $64$ as $8^{2}$ .

$x = \pm \sqrt{8^{2}}$

Step 6.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 8$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 8$

Step 7.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 8$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 8, - 8$