Finite Math Examples

$2805 = (175 - 10 x) (15 + 2 x)$

Step 1

Rewrite the equation as $(175 - 10 x) (15 + 2 x) = 2805$ .

$(175 - 10 x) (15 + 2 x) = 2805$

Step 2

Simplify $(175 - 10 x) (15 + 2 x)$ .

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Step 2.1

Expand $(175 - 10 x) (15 + 2 x)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$175 (15 + 2 x) - 10 x (15 + 2 x) = 2805$

Step 2.1.2

Apply the distributive property.

$175 \cdot 15 + 175 (2 x) - 10 x (15 + 2 x) = 2805$

Step 2.1.3

Apply the distributive property.

$175 \cdot 15 + 175 (2 x) - 10 x \cdot 15 - 10 x (2 x) = 2805$

Step 2.2

Simplify and combine like terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $175$ by $15$ .

$2625 + 175 (2 x) - 10 x \cdot 15 - 10 x (2 x) = 2805$

Step 2.2.1.2

Multiply $2$ by $175$ .

$2625 + 350 x - 10 x \cdot 15 - 10 x (2 x) = 2805$

Step 2.2.1.3

Multiply $15$ by $- 10$ .

$2625 + 350 x - 150 x - 10 x (2 x) = 2805$

Step 2.2.1.4

Rewrite using the commutative property of multiplication.

$2625 + 350 x - 150 x - 10 \cdot 2 x \cdot x = 2805$

Step 2.2.1.5

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

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Step 2.2.1.5.1

Move $x$ .

$2625 + 350 x - 150 x - 10 \cdot 2 (x \cdot x) = 2805$

Step 2.2.1.5.2

Multiply $x$ by $x$ .

$2625 + 350 x - 150 x - 10 \cdot 2 x^{2} = 2805$

Step 2.2.1.6

Multiply $- 10$ by $2$ .

$2625 + 350 x - 150 x - 20 x^{2} = 2805$

Step 2.2.2

Subtract $150 x$ from $350 x$ .

$2625 + 200 x - 20 x^{2} = 2805$

Step 3

Subtract $2805$ from both sides of the equation.

$2625 + 200 x - 20 x^{2} - 2805 = 0$

Step 4

Subtract $2805$ from $2625$ .

$200 x - 20 x^{2} - 180 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $- 20$ out of $200 x - 20 x^{2} - 180$ .

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Step 5.1.1

Reorder $200 x$ and $- 20 x^{2}$ .

$- 20 x^{2} + 200 x - 180 = 0$

Step 5.1.2

Factor $- 20$ out of $- 20 x^{2}$ .

$- 20 x^{2} + 200 x - 180 = 0$

Step 5.1.3

Factor $- 20$ out of $200 x$ .

$- 20 x^{2} - 20 (- 10 x) - 180 = 0$

Step 5.1.4

Factor $- 20$ out of $- 180$ .

$- 20 x^{2} - 20 (- 10 x) - 20 \cdot 9 = 0$

Step 5.1.5

Factor $- 20$ out of $- 20 (x^{2}) - 20 (- 10 x)$ .

$- 20 (x^{2} - 10 x) - 20 \cdot 9 = 0$

Step 5.1.6

Factor $- 20$ out of $- 20 (x^{2} - 10 x) - 20 (9)$ .

$- 20 (x^{2} - 10 x + 9) = 0$

Step 5.2

Factor.

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Step 5.2.1

Factor $x^{2} - 10 x + 9$ using the AC method.

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Step 5.2.1.1

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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Step 5.2.1.2

Write the factored form using these integers.

$- 20 ((x - 9) (x - 1)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$- 20 (x - 9) (x - 1) = 0$

Step 6

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

$x - 9 = 0$

$x - 1 = 0$

Step 7

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 7.2

Add $9$ to both sides of the equation.

$x = 9$

Step 8

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 8.2

Add $1$ to both sides of the equation.

$x = 1$

Step 9

The final solution is all the values that make $- 20 (x - 9) (x - 1) = 0$ true.

$x = 9, 1$