Chemistry Examples

$(40 - 2 x) \cdot (55 - 2 x) = 1350$

Step 1

Simplify $(40 - 2 x) \cdot (55 - 2 x)$ .

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Step 1.1

Expand $(40 - 2 x) (55 - 2 x)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$40 (55 - 2 x) - 2 x (55 - 2 x) = 1350$

Step 1.1.2

Apply the distributive property.

$40 \cdot 55 + 40 (- 2 x) - 2 x (55 - 2 x) = 1350$

Step 1.1.3

Apply the distributive property.

$40 \cdot 55 + 40 (- 2 x) - 2 x \cdot 55 - 2 x (- 2 x) = 1350$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $40$ by $55$ .

$2200 + 40 (- 2 x) - 2 x \cdot 55 - 2 x (- 2 x) = 1350$

Step 1.2.1.2

Multiply $- 2$ by $40$ .

$2200 - 80 x - 2 x \cdot 55 - 2 x (- 2 x) = 1350$

Step 1.2.1.3

Multiply $55$ by $- 2$ .

$2200 - 80 x - 110 x - 2 x (- 2 x) = 1350$

Step 1.2.1.4

Rewrite using the commutative property of multiplication.

$2200 - 80 x - 110 x - 2 \cdot - 2 x \cdot x = 1350$

Step 1.2.1.5

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

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Step 1.2.1.5.1

Move $x$ .

$2200 - 80 x - 110 x - 2 \cdot - 2 (x \cdot x) = 1350$

Step 1.2.1.5.2

Multiply $x$ by $x$ .

$2200 - 80 x - 110 x - 2 \cdot - 2 x^{2} = 1350$

Step 1.2.1.6

Multiply $- 2$ by $- 2$ .

$2200 - 80 x - 110 x + 4 x^{2} = 1350$

Step 1.2.2

Subtract $110 x$ from $- 80 x$ .

$2200 - 190 x + 4 x^{2} = 1350$

Step 2

Subtract $1350$ from both sides of the equation.

$2200 - 190 x + 4 x^{2} - 1350 = 0$

Step 3

Subtract $1350$ from $2200$ .

$- 190 x + 4 x^{2} + 850 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $2$ out of $- 190 x + 4 x^{2} + 850$ .

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Step 4.1.1

Factor $2$ out of $- 190 x$ .

$2 (- 95 x) + 4 x^{2} + 850 = 0$

Step 4.1.2

Factor $2$ out of $4 x^{2}$ .

$2 (- 95 x) + 2 (2 x^{2}) + 850 = 0$

Step 4.1.3

Factor $2$ out of $850$ .

$2 (- 95 x) + 2 (2 x^{2}) + 2 (425) = 0$

Step 4.1.4

Factor $2$ out of $2 (- 95 x) + 2 (2 x^{2})$ .

$2 (- 95 x + 2 x^{2}) + 2 (425) = 0$

Step 4.1.5

Factor $2$ out of $2 (- 95 x + 2 x^{2}) + 2 (425)$ .

$2 (- 95 x + 2 x^{2} + 425) = 0$

Step 4.2

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$2 (- 95 u + 2 u^{2} + 425) = 0$

Step 4.3

Factor by grouping.

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Step 4.3.1

Reorder terms.

$2 (2 u^{2} - 95 u + 425) = 0$

Step 4.3.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 425 = 850$ and whose sum is $b = - 95$ .

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Step 4.3.2.1

Factor $- 95$ out of $- 95 u$ .

$2 (2 u^{2} - 95 u + 425) = 0$

Step 4.3.2.2

Rewrite $- 95$ as $- 10$ plus $- 85$

$2 (2 u^{2} + (- 10 - 85) u + 425) = 0$

Step 4.3.2.3

Apply the distributive property.

$2 (2 u^{2} - 10 u - 85 u + 425) = 0$

Step 4.3.3

Factor out the greatest common factor from each group.

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Step 4.3.3.1

Group the first two terms and the last two terms.

$2 ((2 u^{2} - 10 u) - 85 u + 425) = 0$

Step 4.3.3.2

Factor out the greatest common factor (GCF) from each group.

$2 (2 u (u - 5) - 85 (u - 5)) = 0$

Step 4.3.4

Factor the polynomial by factoring out the greatest common factor, $u - 5$ .

$2 ((u - 5) (2 u - 85)) = 0$

Step 4.4

Factor.

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Step 4.4.1

Replace all occurrences of $u$ with $x$ .

$2 ((x - 5) (2 x - 85)) = 0$

Step 4.4.2

Remove unnecessary parentheses.

$2 (x - 5) (2 x - 85) = 0$

Step 5

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

$x - 5 = 0$

$2 x - 85 = 0$

Step 6

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 6.2

Add $5$ to both sides of the equation.

$x = 5$

Step 7

Set $2 x - 85$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $2 x - 85$ equal to $0$ .

$2 x - 85 = 0$

Step 7.2

Solve $2 x - 85 = 0$ for $x$ .

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Step 7.2.1

Add $85$ to both sides of the equation.

$2 x = 85$

Step 7.2.2

Divide each term in $2 x = 85$ by $2$ and simplify.

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Step 7.2.2.1

Divide each term in $2 x = 85$ by $2$ .

$\frac{2 x}{2} = \frac{85}{2}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{85}{2}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{85}{2}$

Step 8

The final solution is all the values that make $2 (x - 5) (2 x - 85) = 0$ true.

$x = 5, \frac{85}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 5, \frac{85}{2}$

Decimal Form:

$x = 5, 42.5$

Mixed Number Form:

$x = 5, 42 \frac{1}{2}$