Algebra Examples

$49 x^{4} - 4 {(x^{2} - 3 x)}^{2}$

Step 1

Rewrite $49 x^{4}$ as ${(7 x^{2})}^{2}$ .

${(7 x^{2})}^{2} - 4 {(x^{2} - 3 x)}^{2}$

Step 2

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

${(7 x^{2})}^{2} - {(2 (x^{2} - 3 x))}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7 x^{2}$ and $b = 2 (x^{2} - 3 x)$ .

$(7 x^{2} + 2 (x^{2} - 3 x)) (7 x^{2} - (2 (x^{2} - 3 x)))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(7 x^{2} + 2 x^{2} + 2 (- 3 x)) (7 x^{2} - (2 (x^{2} - 3 x)))$

Step 4.2

Multiply $- 3$ by $2$ .

$(7 x^{2} + 2 x^{2} - 6 x) (7 x^{2} - (2 (x^{2} - 3 x)))$

Step 4.3

Add $7 x^{2}$ and $2 x^{2}$ .

$(9 x^{2} - 6 x) (7 x^{2} - (2 (x^{2} - 3 x)))$

Step 4.4

Factor $3 x$ out of $9 x^{2} - 6 x$ .

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Step 4.4.1

Factor $3 x$ out of $9 x^{2}$ .

$(3 x (3 x) - 6 x) (7 x^{2} - (2 (x^{2} - 3 x)))$

Step 4.4.2

Factor $3 x$ out of $- 6 x$ .

$(3 x (3 x) + 3 x (- 2)) (7 x^{2} - (2 (x^{2} - 3 x)))$

Step 4.4.3

Factor $3 x$ out of $3 x (3 x) + 3 x (- 2)$ .

$3 x (3 x - 2) (7 x^{2} - (2 (x^{2} - 3 x)))$

Step 4.5

Multiply $2$ by $- 1$ .

$3 x (3 x - 2) (7 x^{2} - 2 (x^{2} - 3 x))$

Step 5

Apply the distributive property.

$3 x (3 x - 2) (7 x^{2} - 2 x^{2} - 2 (- 3 x))$

Step 6

Multiply $- 3$ by $- 2$ .

$3 x (3 x - 2) (7 x^{2} - 2 x^{2} + 6 x)$

Step 7

Subtract $2 x^{2}$ from $7 x^{2}$ .

$3 x (3 x - 2) (5 x^{2} + 6 x)$

Step 8

Factor.

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Step 8.1

Factor $x$ out of $5 x^{2} + 6 x$ .

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Step 8.1.1

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

$3 x (3 x - 2) (x (5 x) + 6 x)$

Step 8.1.2

Factor $x$ out of $6 x$ .

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

Step 8.1.3

Factor $x$ out of $x (5 x) + x \cdot 6$ .

$3 x (3 x - 2) (x (5 x + 6))$

Step 8.2

Remove unnecessary parentheses.

$3 x (3 x - 2) x (5 x + 6)$

Step 9

Combine exponents.

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Step 9.1

Raise $x$ to the power of $1$ .

$3 (x^{1} x) (3 x - 2) (5 x + 6)$

Step 9.2

Raise $x$ to the power of $1$ .

$3 (x^{1} x^{1}) (3 x - 2) (5 x + 6)$

Step 9.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{1 + 1} (3 x - 2) (5 x + 6)$

Step 9.4

Add $1$ and $1$ .

$3 x^{2} (3 x - 2) (5 x + 6)$