Pre-Algebra Examples

$1750 = x (85 - x)$

Step 1

Rewrite the equation as $x (85 - x) = 1750$ .

$x (85 - x) = 1750$

Step 2

Simplify $x (85 - x)$ .

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Step 2.1

Simplify by multiplying through.

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Step 2.1.1

Apply the distributive property.

$x \cdot 85 + x (- x) = 1750$

Step 2.1.2

Reorder.

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Step 2.1.2.1

Move $85$ to the left of $x$ .

$85 \cdot x + x (- x) = 1750$

Step 2.1.2.2

Rewrite using the commutative property of multiplication.

$85 \cdot x - x \cdot x = 1750$

Step 2.2

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

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Step 2.2.1

Move $x$ .

$85 x - (x \cdot x) = 1750$

Step 2.2.2

Multiply $x$ by $x$ .

$85 x - x^{2} = 1750$

Step 3

Subtract $1750$ from both sides of the equation.

$85 x - x^{2} - 1750 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $- 1$ out of $85 x - x^{2} - 1750$ .

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Step 4.1.1

Reorder $85 x$ and $- x^{2}$ .

$- x^{2} + 85 x - 1750 = 0$

Step 4.1.2

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

$- (x^{2}) + 85 x - 1750 = 0$

Step 4.1.3

Factor $- 1$ out of $85 x$ .

$- (x^{2}) - (- 85 x) - 1750 = 0$

Step 4.1.4

Rewrite $- 1750$ as $- 1 (1750)$ .

$- (x^{2}) - (- 85 x) - 1 \cdot 1750 = 0$

Step 4.1.5

Factor $- 1$ out of $- (x^{2}) - (- 85 x)$ .

$- (x^{2} - 85 x) - 1 \cdot 1750 = 0$

Step 4.1.6

Factor $- 1$ out of $- (x^{2} - 85 x) - 1 (1750)$ .

$- (x^{2} - 85 x + 1750) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $x^{2} - 85 x + 1750$ using the AC method.

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Step 4.2.1.1

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 $1750$ and whose sum is $- 85$ .

$- 50, - 35$

Step 4.2.1.2

Write the factored form using these integers.

$- ((x - 50) (x - 35)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$- (x - 50) (x - 35) = 0$

Step 5

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

$x - 50 = 0$

$x - 35 = 0$

Step 6

Set $x - 50$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 50$ equal to $0$ .

$x - 50 = 0$

Step 6.2

Add $50$ to both sides of the equation.

$x = 50$

Step 7

Set $x - 35$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 35$ equal to $0$ .

$x - 35 = 0$

Step 7.2

Add $35$ to both sides of the equation.

$x = 35$

Step 8

The final solution is all the values that make $- (x - 50) (x - 35) = 0$ true.

$x = 50, 35$