Pre-Algebra Examples

$x \cdot (2 x + 200) = 52800$

Step 1

Simplify $x \cdot (2 x + 200)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$x (2 x) + x \cdot 200 = 52800$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot 200 = 52800$

Step 1.1.2.2

Move $200$ to the left of $x$ .

$2 x \cdot x + 200 \cdot x = 52800$

Step 1.2

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

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Step 1.2.1

Move $x$ .

$2 (x \cdot x) + 200 \cdot x = 52800$

Step 1.2.2

Multiply $x$ by $x$ .

$2 x^{2} + 200 \cdot x = 52800$

$2 x^{2} + 200 x = 52800$

Step 2

Subtract $52800$ from both sides of the equation.

$2 x^{2} + 200 x - 52800 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $2$ out of $2 x^{2} + 200 x - 52800$ .

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Step 3.1.1

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

$2 (x^{2}) + 200 x - 52800 = 0$

Step 3.1.2

Factor $2$ out of $200 x$ .

$2 (x^{2}) + 2 (100 x) - 52800 = 0$

Step 3.1.3

Factor $2$ out of $- 52800$ .

$2 x^{2} + 2 (100 x) + 2 \cdot - 26400 = 0$

Step 3.1.4

Factor $2$ out of $2 x^{2} + 2 (100 x)$ .

$2 (x^{2} + 100 x) + 2 \cdot - 26400 = 0$

Step 3.1.5

Factor $2$ out of $2 (x^{2} + 100 x) + 2 \cdot - 26400$ .

$2 (x^{2} + 100 x - 26400) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $x^{2} + 100 x - 26400$ using the AC method.

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Step 3.2.1.1

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 $- 26400$ and whose sum is $100$ .

$- 120, 220$

Step 3.2.1.2

Write the factored form using these integers.

$2 ((x - 120) (x + 220)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$2 (x - 120) (x + 220) = 0$

Step 4

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

$x - 120 = 0$

$x + 220 = 0$

Step 5

Set $x - 120$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 120$ equal to $0$ .

$x - 120 = 0$

Step 5.2

Add $120$ to both sides of the equation.

$x = 120$

Step 6

Set $x + 220$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 220$ equal to $0$ .

$x + 220 = 0$

Step 6.2

Subtract $220$ from both sides of the equation.

$x = - 220$

Step 7

The final solution is all the values that make $2 (x - 120) (x + 220) = 0$ true.

$x = 120, - 220$