Basic Math Examples

$y (11 - y) = 30$

Step 1

Simplify $y (11 - y)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$y \cdot 11 + y (- y) = 30$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Move $11$ to the left of $y$ .

$11 \cdot y + y (- y) = 30$

Step 1.1.2.2

Rewrite using the commutative property of multiplication.

$11 \cdot y - y \cdot y = 30$

Step 1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 1.2.1

Move $y$ .

$11 y - (y \cdot y) = 30$

Step 1.2.2

Multiply $y$ by $y$ .

$11 y - y^{2} = 30$

Step 2

Subtract $30$ from both sides of the equation.

$11 y - y^{2} - 30 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 1$ out of $11 y - y^{2} - 30$ .

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Step 3.1.1

Reorder $11 y$ and $- y^{2}$ .

$- y^{2} + 11 y - 30 = 0$

Step 3.1.2

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

$- (y^{2}) + 11 y - 30 = 0$

Step 3.1.3

Factor $- 1$ out of $11 y$ .

$- (y^{2}) - (- 11 y) - 30 = 0$

Step 3.1.4

Rewrite $- 30$ as $- 1 (30)$ .

$- (y^{2}) - (- 11 y) - 1 \cdot 30 = 0$

Step 3.1.5

Factor $- 1$ out of $- (y^{2}) - (- 11 y)$ .

$- (y^{2} - 11 y) - 1 \cdot 30 = 0$

Step 3.1.6

Factor $- 1$ out of $- (y^{2} - 11 y) - 1 (30)$ .

$- (y^{2} - 11 y + 30) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $y^{2} - 11 y + 30$ 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 $30$ and whose sum is $- 11$ .

$- 6, - 5$

Step 3.2.1.2

Write the factored form using these integers.

$- ((y - 6) (y - 5)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$- (y - 6) (y - 5) = 0$

Step 4

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

$y - 6 = 0$

$y - 5 = 0$

Step 5

Set $y - 6$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

Step 5.2

Add $6$ to both sides of the equation.

$y = 6$

Step 6

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

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Step 6.1

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

$y - 5 = 0$

Step 6.2

Add $5$ to both sides of the equation.

$y = 5$

Step 7

The final solution is all the values that make $- (y - 6) (y - 5) = 0$ true.

$y = 6, 5$