Basic Math Examples

$\frac{5 y^{2} + 3}{y^{2} - 2 y} + \frac{10 y + 3}{2 y - y^{2}}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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Step 1.1.1.1

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

$\frac{5 y^{2} + 3}{y \cdot y - 2 y} + \frac{10 y + 3}{2 y - y^{2}}$

Step 1.1.1.2

Factor $y$ out of $- 2 y$ .

$\frac{5 y^{2} + 3}{y \cdot y + y \cdot - 2} + \frac{10 y + 3}{2 y - y^{2}}$

Step 1.1.1.3

Factor $y$ out of $y \cdot y + y \cdot - 2$ .

$\frac{5 y^{2} + 3}{y (y - 2)} + \frac{10 y + 3}{2 y - y^{2}}$

Step 1.1.2

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

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Step 1.1.2.1

Factor $y$ out of $2 y$ .

$\frac{5 y^{2} + 3}{y (y - 2)} + \frac{10 y + 3}{y \cdot 2 - y^{2}}$

Step 1.1.2.2

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

$\frac{5 y^{2} + 3}{y (y - 2)} + \frac{10 y + 3}{y \cdot 2 + y (- y)}$

Step 1.1.2.3

Factor $y$ out of $y \cdot 2 + y (- y)$ .

$\frac{5 y^{2} + 3}{y (y - 2)} + \frac{10 y + 3}{y (2 - y)}$

Step 1.2

Simplify terms.

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Step 1.2.1

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{5 y^{2} + 3}{y (y - 2)} + \frac{10 y + 3}{y (- 1 (- 2) - y)}$

Step 1.2.2

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

$\frac{5 y^{2} + 3}{y (y - 2)} + \frac{10 y + 3}{y (- 1 (- 2) - (y))}$

Step 1.2.3

Factor $- 1$ out of $- 1 (- 2) - (y)$ .

$\frac{5 y^{2} + 3}{y (y - 2)} + \frac{10 y + 3}{y (- 1 (- 2 + y))}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Move a negative from the denominator of $\frac{10 y + 3}{y (- 1 (- 2 + y))}$ to the numerator.

$\frac{5 y^{2} + 3}{y (y - 2)} + \frac{- (10 y + 3)}{y (- 2 + y)}$

Step 1.2.4.2

Reorder terms.

$\frac{5 y^{2} + 3}{y (y - 2)} + \frac{- (10 y + 3)}{y (y - 2)}$

Step 1.2.5

Combine the numerators over the common denominator.

$\frac{5 y^{2} + 3 - (10 y + 3)}{y (y - 2)}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the distributive property.

$\frac{5 y^{2} + 3 - (10 y) - 1 \cdot 3}{y (y - 2)}$

Step 2.2

Multiply $10$ by $- 1$ .

$\frac{5 y^{2} + 3 - 10 y - 1 \cdot 3}{y (y - 2)}$

Step 2.3

Multiply $- 1$ by $3$ .

$\frac{5 y^{2} + 3 - 10 y - 3}{y (y - 2)}$

Step 2.4

Subtract $3$ from $3$ .

$\frac{5 y^{2} - 10 y + 0}{y (y - 2)}$

Step 2.5

Add $5 y^{2} - 10 y$ and $0$ .

$\frac{5 y^{2} - 10 y}{y (y - 2)}$

Step 2.6

Factor $5 y$ out of $5 y^{2} - 10 y$ .

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Step 2.6.1

Factor $5 y$ out of $5 y^{2}$ .

$\frac{5 y (y) - 10 y}{y (y - 2)}$

Step 2.6.2

Factor $5 y$ out of $- 10 y$ .

$\frac{5 y (y) + 5 y (- 2)}{y (y - 2)}$

Step 2.6.3

Factor $5 y$ out of $5 y (y) + 5 y (- 2)$ .

$\frac{5 y (y - 2)}{y (y - 2)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $y$ .

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Step 3.1.1

Cancel the common factor.

$\frac{5 y (y - 2)}{y (y - 2)}$

Step 3.1.2

Rewrite the expression.

$\frac{5 (y - 2)}{y - 2}$

Step 3.2

Cancel the common factor of $y - 2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{5 (y - 2)}{y - 2}$

Step 3.2.2

Divide $5$ by $1$ .

$5$