factor -36a^2+9a^4 — step-by-step solution

1) Let's factor in

ℝ

- 36 a^{2} + 9 a^{4}

We factor out

9

, the greatest common divisor of the coefficients:

- 36 a^{2} + 9 a^{4}

9 \cdot (- 4 a^{2} + a^{4})

For

- 4 a^{2} + a^{4}

, we recognize a difference of two squares:

{(a^{2})}^{2} - {(2 a)}^{2}

. Apply

A^{2} - B^{2} = (A - B) \cdot (A + B)

, with

A = a^{2}

and

B = 2 a

You can write the expression:

- 4 a^{2} + a^{4}

in factored form:

(a^{2} - 2 a) \cdot (a^{2} + 2 a)

2) Let's factorize in

ℝ

(a^{2} - 2 a) \cdot (a^{2} + 2 a)

⇥2.1) Let's factor in

ℝ

(a^{2} - 2 a)

⇥You can factorize this expression into:

a \cdot (a - 2)

▷The factor

(a^{2} - 2 a)

factors as:

a \cdot (a - 2)

⇥2.2) Let's factor in

ℝ

(a^{2} + 2 a)

⇥You can factorize this expression into:

a \cdot (a + 2)

▷The factor

(a^{2} + 2 a)

factors as:

a \cdot (a + 2)

3) Let's replace each factor by its factorization:

a \cdot (a - 2) \cdot a \cdot (a + 2)

ⓘ Multiplying powers with the same base adds the exponents:

a \cdot a

a^{2}

a^{2} \cdot (a - 2) \cdot (a + 2)

ⓘ Factor out the common factor

a^{2}

▶Factored expression:

9 a^{2} \cdot (a + 2) \cdot (a - 2)