Basic Math Examples

$16 y^{2} - 49 + 3 {(4 y + 7)}^{2}$

Step 1

Rewrite ${(4 y + 7)}^{2}$ as $(4 y + 7) (4 y + 7)$ .

$16 y^{2} - 49 + 3 ((4 y + 7) (4 y + 7))$

Step 2

Expand $(4 y + 7) (4 y + 7)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$16 y^{2} - 49 + 3 (4 y (4 y + 7) + 7 (4 y + 7))$

Step 2.2

Apply the distributive property.

$16 y^{2} - 49 + 3 (4 y (4 y) + 4 y \cdot 7 + 7 (4 y + 7))$

Step 2.3

Apply the distributive property.

$16 y^{2} - 49 + 3 (4 y (4 y) + 4 y \cdot 7 + 7 (4 y) + 7 \cdot 7)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$16 y^{2} - 49 + 3 (4 \cdot 4 y \cdot y + 4 y \cdot 7 + 7 (4 y) + 7 \cdot 7)$

Step 3.1.2

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

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Step 3.1.2.1

Move $y$ .

$16 y^{2} - 49 + 3 (4 \cdot 4 (y \cdot y) + 4 y \cdot 7 + 7 (4 y) + 7 \cdot 7)$

Step 3.1.2.2

Multiply $y$ by $y$ .

$16 y^{2} - 49 + 3 (4 \cdot 4 y^{2} + 4 y \cdot 7 + 7 (4 y) + 7 \cdot 7)$

Step 3.1.3

Multiply $4$ by $4$ .

$16 y^{2} - 49 + 3 (16 y^{2} + 4 y \cdot 7 + 7 (4 y) + 7 \cdot 7)$

Step 3.1.4

Multiply $7$ by $4$ .

$16 y^{2} - 49 + 3 (16 y^{2} + 28 y + 7 (4 y) + 7 \cdot 7)$

Step 3.1.5

Multiply $4$ by $7$ .

$16 y^{2} - 49 + 3 (16 y^{2} + 28 y + 28 y + 7 \cdot 7)$

Step 3.1.6

Multiply $7$ by $7$ .

$16 y^{2} - 49 + 3 (16 y^{2} + 28 y + 28 y + 49)$

Step 3.2

Add $28 y$ and $28 y$ .

$16 y^{2} - 49 + 3 (16 y^{2} + 56 y + 49)$

Step 4

Apply the distributive property.

$16 y^{2} - 49 + 3 (16 y^{2}) + 3 (56 y) + 3 \cdot 49$

Step 5

Simplify.

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Step 5.1

Multiply $16$ by $3$ .

$16 y^{2} - 49 + 48 y^{2} + 3 (56 y) + 3 \cdot 49$

Step 5.2

Multiply $56$ by $3$ .

$16 y^{2} - 49 + 48 y^{2} + 168 y + 3 \cdot 49$

Step 5.3

Multiply $3$ by $49$ .

$16 y^{2} - 49 + 48 y^{2} + 168 y + 147$

Step 6

Add $16 y^{2}$ and $48 y^{2}$ .

$64 y^{2} - 49 + 168 y + 147$

Step 7

Add $- 49$ and $147$ .

$64 y^{2} + 168 y + 98$

Step 8

Rewrite $64 y^{2} + 168 y + 98$ in a factored form.

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Step 8.1

Factor $2$ out of $64 y^{2} + 168 y + 98$ .

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Step 8.1.1

Factor $2$ out of $64 y^{2}$ .

$2 (32 y^{2}) + 168 y + 98$

Step 8.1.2

Factor $2$ out of $168 y$ .

$2 (32 y^{2}) + 2 (84 y) + 98$

Step 8.1.3

Factor $2$ out of $98$ .

$2 (32 y^{2}) + 2 (84 y) + 2 (49)$

Step 8.1.4

Factor $2$ out of $2 (32 y^{2}) + 2 (84 y)$ .

$2 (32 y^{2} + 84 y) + 2 (49)$

Step 8.1.5

Factor $2$ out of $2 (32 y^{2} + 84 y) + 2 (49)$ .

$2 (32 y^{2} + 84 y + 49)$

Step 8.2

Factor.

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Step 8.2.1

Factor by grouping.

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Step 8.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 32 \cdot 49 = 1568$ and whose sum is $b = 84$ .

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Step 8.2.1.1.1

Factor $84$ out of $84 y$ .

$2 (32 y^{2} + 84 (y) + 49)$

Step 8.2.1.1.2

Rewrite $84$ as $28$ plus $56$

$2 (32 y^{2} + (28 + 56) y + 49)$

Step 8.2.1.1.3

Apply the distributive property.

$2 (32 y^{2} + 28 y + 56 y + 49)$

Step 8.2.1.2

Factor out the greatest common factor from each group.

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Step 8.2.1.2.1

Group the first two terms and the last two terms.

$2 ((32 y^{2} + 28 y) + 56 y + 49)$

Step 8.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$2 (4 y (8 y + 7) + 7 (8 y + 7))$

Step 8.2.1.3

Factor the polynomial by factoring out the greatest common factor, $8 y + 7$ .

$2 ((8 y + 7) (4 y + 7))$

Step 8.2.2

Remove unnecessary parentheses.

$2 (8 y + 7) (4 y + 7)$