Pre-Algebra Examples

$x (27 - x) = 170$

Step 1

Simplify $x (27 - x)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$x \cdot 27 + x (- x) = 170$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Move $27$ to the left of $x$ .

$27 \cdot x + x (- x) = 170$

Step 1.1.2.2

Rewrite using the commutative property of multiplication.

$27 \cdot x - x \cdot x = 170$

Step 1.2

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

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Step 1.2.1

Move $x$ .

$27 x - (x \cdot x) = 170$

Step 1.2.2

Multiply $x$ by $x$ .

$27 x - x^{2} = 170$

Step 2

Subtract $170$ from both sides of the equation.

$27 x - x^{2} - 170 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 1$ out of $27 x - x^{2} - 170$ .

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Step 3.1.1

Reorder $27 x$ and $- x^{2}$ .

$- x^{2} + 27 x - 170 = 0$

Step 3.1.2

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

$- (x^{2}) + 27 x - 170 = 0$

Step 3.1.3

Factor $- 1$ out of $27 x$ .

$- (x^{2}) - (- 27 x) - 170 = 0$

Step 3.1.4

Rewrite $- 170$ as $- 1 (170)$ .

$- (x^{2}) - (- 27 x) - 1 \cdot 170 = 0$

Step 3.1.5

Factor $- 1$ out of $- (x^{2}) - (- 27 x)$ .

$- (x^{2} - 27 x) - 1 \cdot 170 = 0$

Step 3.1.6

Factor $- 1$ out of $- (x^{2} - 27 x) - 1 (170)$ .

$- (x^{2} - 27 x + 170) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $x^{2} - 27 x + 170$ 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 $170$ and whose sum is $- 27$ .

$- 17, - 10$

Step 3.2.1.2

Write the factored form using these integers.

$- ((x - 17) (x - 10)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$- (x - 17) (x - 10) = 0$

Step 4

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

$x - 17 = 0$

$x - 10 = 0$

Step 5

Set $x - 17$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 17$ equal to $0$ .

$x - 17 = 0$

Step 5.2

Add $17$ to both sides of the equation.

$x = 17$

Step 6

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 6.2

Add $10$ to both sides of the equation.

$x = 10$

Step 7

The final solution is all the values that make $- (x - 17) (x - 10) = 0$ true.

$x = 17, 10$