factor x^2-y^2-4x+4 — step-by-step solution

The expression:

x^{2}

- y^{2}

- 4 x

+ 4

contains several variables so we pick x for factorization

Let's factor in

ℝ

x^{2}

- y^{2}

- 4 x

+ 4

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -4

and

c = - y^{2} + 4

Δ = {(-4)}^{2} - 4 \cdot 1 \cdot (- y^{2} + 4)

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-4)}^{2}

(-4) \cdot (-4)

16

Δ = 16 - 4 \cdot (- y^{2} + 4)

ⓘ Expand by multiplying each term inside the parentheses:

- 4 \cdot (- y^{2} + 4)

- (4 \cdot (- y^{2}) + 16)

Δ = 16 - (4 \cdot (- y^{2}) + 16)

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (4 \cdot (- y^{2}) + 16)

- 4 \cdot (- y^{2}) - 16

Δ = 16 - 4 \cdot (- y^{2}) - 16

ⓘ Two minus signs cancel out:

-4 \cdot (- y^{2})

4 y^{2}

■Start from

16

and move back

16

on the number line, down to zero:

16 - 16

0

Δ = 0 + 4 y^{2}

ⓘ Adding 0 changes nothing:

0 + 4 y^{2}

4 y^{2}

Δ = 4 y^{2}

The discriminant is a square:

Δ = {(2 y)}^{2}

. With

q = 2 y

, the roots are

\frac{- b - q}{2 a}

and

\frac{- b + q}{2 a}

, including when q is 0

x1 = \frac{4 - 2 y}{2}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{4 - 2 y}{2}

\frac{2 \cdot (2 - y)}{2}

2 - y

x1 = 2 - y

x2 = \frac{4 + 2 y}{2}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{4 + 2 y}{2}

\frac{2 \cdot (2 + y)}{2}

2 + y

x2 = 2 + y

Factors found:

(x - (y + 2)) \cdot (x - (2 - y))

(x - (2 - y)) \cdot (x - (y + 2))

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (2 - y)

- 2 + y

- (y + 2)

- y - 2

(x - 2 + y) \cdot (x - y - 2)

▶Factored expression:

(x + y - 2) \cdot (x - y - 2)