factor x^2-225y^2 — step-by-step solution

The expression

x^{2} - 225 y^{2}

contains several variables: the variable x is chosen for factorization

Let's factor in

ℝ

x^{2} - 225 y^{2}

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 = 0

and

c = - 225 y^{2}

Δ = 0^{2} - 4 \cdot 1 \cdot (- 225 y^{2})

ⓘ Adding 0 changes nothing:

0 - 4 \cdot (-225) y^{2}

- 4 \cdot (-225) y^{2}

Δ = - 4 \cdot (-225) y^{2}

ⓘ Two minus signs cancel out, the product of two negative numbers is positive:

-4 \cdot (-225)

4 \cdot 225

4 \cdot (200 + 20 + 5)

4 \cdot 200 + 4 \cdot 20 + 4 \cdot 5

800 + 80 + 20

900

Δ = 900 y^{2}

The discriminant is a square:

Δ = {(30 y)}^{2}

. With

q = 30 y

, the roots are

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

and

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

, including when q is 0

x1 = - \frac{30 y}{2}

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

- \frac{30 y}{2}

- 15 y

x1 = - 15 y

x2 = \frac{30 y}{2}

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

\frac{30 y}{2}

15 y

x2 = 15 y

▶Factors identified:

(x + 15 y) \cdot (x - 15 y)