Finite Math Examples

$200 x - 2 x^{2} = - x^{2} + 25 x + 6900$

Step 1

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

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Step 1.1

Add $x^{2}$ to both sides of the equation.

$200 x - 2 x^{2} + x^{2} = 25 x + 6900$

Step 1.2

Subtract $25 x$ from both sides of the equation.

$200 x - 2 x^{2} + x^{2} - 25 x = 6900$

Step 1.3

Subtract $25 x$ from $200 x$ .

$- 2 x^{2} + x^{2} + 175 x = 6900$

Step 1.4

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

$- x^{2} + 175 x = 6900$

Step 2

Subtract $6900$ from both sides of the equation.

$- x^{2} + 175 x - 6900 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 1$ out of $- x^{2} + 175 x - 6900$ .

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Step 3.1.1

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 175 x - 6900 = 0$

Step 3.1.2

Factor $- 1$ out of $175 x$ .

$- (x^{2}) - (- 175 x) - 6900 = 0$

Step 3.1.3

Rewrite $- 6900$ as $- 1 (6900)$ .

$- (x^{2}) - (- 175 x) - 1 \cdot 6900 = 0$

Step 3.1.4

Factor $- 1$ out of $- (x^{2}) - (- 175 x)$ .

$- (x^{2} - 175 x) - 1 \cdot 6900 = 0$

Step 3.1.5

Factor $- 1$ out of $- (x^{2} - 175 x) - 1 (6900)$ .

$- (x^{2} - 175 x + 6900) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $x^{2} - 175 x + 6900$ 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 $6900$ and whose sum is $- 175$ .

$- 115, - 60$

Step 3.2.1.2

Write the factored form using these integers.

$- ((x - 115) (x - 60)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$- (x - 115) (x - 60) = 0$

Step 4

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

$x - 115 = 0$

$x - 60 = 0$

Step 5

Set $x - 115$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 115$ equal to $0$ .

$x - 115 = 0$

Step 5.2

Add $115$ to both sides of the equation.

$x = 115$

Step 6

Set $x - 60$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 60$ equal to $0$ .

$x - 60 = 0$

Step 6.2

Add $60$ to both sides of the equation.

$x = 60$

Step 7

The final solution is all the values that make $- (x - 115) (x - 60) = 0$ true.

$x = 115, 60$