Algebra Examples

$25 {(4 x - 3)}^{2} - 9 {(2 x + 1)}^{2}$

Step 1

Rewrite $25 {(4 x - 3)}^{2}$ as ${(5 (4 x - 3))}^{2}$ .

${(5 (4 x - 3))}^{2} - 9 {(2 x + 1)}^{2}$

Step 2

Rewrite $9 {(2 x + 1)}^{2}$ as ${(3 (2 x + 1))}^{2}$ .

${(5 (4 x - 3))}^{2} - {(3 (2 x + 1))}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 (4 x - 3)$ and $b = 3 (2 x + 1)$ .

$(5 (4 x - 3) + 3 (2 x + 1)) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(5 (4 x) + 5 \cdot - 3 + 3 (2 x + 1)) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.2

Multiply $4$ by $5$ .

$(20 x + 5 \cdot - 3 + 3 (2 x + 1)) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.3

Multiply $5$ by $- 3$ .

$(20 x - 15 + 3 (2 x + 1)) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.4

Apply the distributive property.

$(20 x - 15 + 3 (2 x) + 3 \cdot 1) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.5

Multiply $2$ by $3$ .

$(20 x - 15 + 6 x + 3 \cdot 1) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.6

Multiply $3$ by $1$ .

$(20 x - 15 + 6 x + 3) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.7

Add $20 x$ and $6 x$ .

$(26 x - 15 + 3) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.8

Add $- 15$ and $3$ .

$(26 x - 12) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.9

Factor $2$ out of $26 x - 12$ .

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Step 4.9.1

Factor $2$ out of $26 x$ .

$(2 (13 x) - 12) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.9.2

Factor $2$ out of $- 12$ .

$(2 (13 x) + 2 (- 6)) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.9.3

Factor $2$ out of $2 (13 x) + 2 (- 6)$ .

$2 (13 x - 6) (5 (4 x - 3) - (3 (2 x + 1)))$

Step 4.10

Multiply $3$ by $- 1$ .

$2 (13 x - 6) (5 (4 x - 3) - 3 (2 x + 1))$

Step 5

Apply the distributive property.

$2 (13 x - 6) (5 (4 x) + 5 \cdot - 3 - 3 (2 x + 1))$

Step 6

Multiply $4$ by $5$ .

$2 (13 x - 6) (20 x + 5 \cdot - 3 - 3 (2 x + 1))$

Step 7

Multiply $5$ by $- 3$ .

$2 (13 x - 6) (20 x - 15 - 3 (2 x + 1))$

Step 8

Apply the distributive property.

$2 (13 x - 6) (20 x - 15 - 3 (2 x) - 3 \cdot 1)$

Step 9

Multiply $2$ by $- 3$ .

$2 (13 x - 6) (20 x - 15 - 6 x - 3 \cdot 1)$

Step 10

Multiply $- 3$ by $1$ .

$2 (13 x - 6) (20 x - 15 - 6 x - 3)$

Step 11

Subtract $6 x$ from $20 x$ .

$2 (13 x - 6) (14 x - 15 - 3)$

Step 12

Subtract $3$ from $- 15$ .

$2 (13 x - 6) (14 x - 18)$

Step 13

Factor.

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Step 13.1

Factor $2$ out of $14 x - 18$ .

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Step 13.1.1

Factor $2$ out of $14 x$ .

$2 (13 x - 6) (2 (7 x) - 18)$

Step 13.1.2

Factor $2$ out of $- 18$ .

$2 (13 x - 6) (2 (7 x) + 2 (- 9))$

Step 13.1.3

Factor $2$ out of $2 (7 x) + 2 (- 9)$ .

$2 (13 x - 6) (2 (7 x - 9))$

Step 13.2

Remove unnecessary parentheses.

$2 (13 x - 6) \cdot 2 (7 x - 9)$

Step 14

Multiply $2$ by $2$ .

$4 (13 x - 6) (7 x - 9)$