Basic Math Examples

$y^{2} - 4 z^{2} - 6 y + 9$

Step 1

Regroup terms.

$- 4 z^{2} + y^{2} - 6 y + 9$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$- 4 z^{2} + y^{2} - 6 y + 3^{2}$

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 y = 2 \cdot y \cdot 3$

Step 2.3

Rewrite the polynomial.

$- 4 z^{2} + y^{2} - 2 \cdot y \cdot 3 + 3^{2}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 3$ .

$- 4 z^{2} + {(y - 3)}^{2}$

Step 3

Rewrite $4 z^{2}$ as ${(2 z)}^{2}$ .

$- {(2 z)}^{2} + {(y - 3)}^{2}$

Step 4

Reorder $- {(2 z)}^{2}$ and ${(y - 3)}^{2}$ .

${(y - 3)}^{2} - {(2 z)}^{2}$

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y - 3$ and $b = 2 z$ .

$(y - 3 + 2 z) (y - 3 - (2 z))$

Step 6

Multiply $2$ by $- 1$ .

$(y - 3 + 2 z) (y - 3 - 2 z)$