Math Snap

STEP 1

1. We are asked to factorize two algebraic expressions.
2. The expressions can be simplified using algebraic identities and factoring techniques.

STEP 2

1. Factorize the expression $A = 4x^2 - (x-1)^2$.
2. Factorize the expression $B = x^2 - 27$.

STEP 3

Expand the expression $(x-1)^2$ in $A = 4x^2 - (x-1)^2$.
$$(x-1)^2 = x^2 - 2x + 1$$ Substitute back into the expression for $A$:
$$A = 4x^2 - (x^2 - 2x + 1)$$

STEP 4

Simplify the expression by distributing the negative sign:
$$A = 4x^2 - x^2 + 2x - 1$$ Combine like terms:
$$A = 3x^2 + 2x - 1$$

STEP 5

Factor the quadratic expression $3x^2 + 2x - 1$.
To factor, find two numbers that multiply to $3 \times (-1) = -3$ and add to $2$. These numbers are $3$ and $-1$.
Rewrite the middle term using these numbers:
$$3x^2 + 3x - x - 1$$ Group terms:
$$(3x^2 + 3x) + (-x - 1)$$ Factor by grouping:
$$3x(x + 1) - 1(x + 1)$$ Factor out the common factor $(x + 1)$:
$$A = (3x - 1)(x + 1)$$

Factor the expression $B = x^2 - 27$.
Recognize this as a difference of squares:
$$B = x^2 - (\sqrt{27})^2$$ Use the difference of squares formula $a^2 - b^2 = (a - b)(a + b)$:
$$B = (x - \sqrt{27})(x + \sqrt{27})$$ Simplify $\sqrt{27}$ to $3\sqrt{3}$:
$$B = (x - 3\sqrt{3})(x + 3\sqrt{3})$$