simplify (a^2-1)/(a^2+2a-3) — step-by-step solution

The original denominators must be nonzero:

a \in ℝ \ {-3, 1}

1) Let's reduce:

\frac{a^{2} - 1}{a^{2} + 2 a - 3}

2) Let's factor in

ℝ

a^{2} - 1

For

a^{2} - 1

, we recognize a difference of two squares:

a^{2} - 1^{2}

. Apply

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

, with

A = a

and

B = 1

Factors found:

(a + 1) \cdot (a - 1)

3) Let's factor in

ℝ

a^{2} + 2 a - 3

Given the quadratic form

A a^{2} + B a + C

, let's compute the discriminant:

Δ = B^{2} - 4 A C

. Here

A = 1

B = 2

and

C = -3

Δ = 2^{2} - 4 \cdot 1 \cdot (-3)

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

2^{2}

2 \cdot 2

4

Δ = 4 - 4 \cdot (-3)

ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-3)

12

Δ = 4 + 12

ⓘ Start from

4

and move forward

12

on the number line:

4 + 12

16

Δ = 16

Δ > 0

, the equation has two roots:

The quadratic formula gives

a_{1}

\frac{- B - \sqrt{Δ}}{2 A}

\frac{- 2 - \sqrt{16}}{2 \cdot 1}

a_{1} = \frac{- 2 - \sqrt{16}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{16}

4

;

4^{2}

16

a_{1} = \frac{- 2 - 4}{2}

ⓘ The numerator and the denominator are computed before the division:

- 2 - 4

-6

a_{1} = - \frac{6}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{6}{2}

-3

a_{1} = -3

The quadratic formula gives

a_{2}

\frac{- B + \sqrt{Δ}}{2 A}

\frac{- 2 + \sqrt{16}}{2 \cdot 1}

a_{2} = \frac{- 2 + \sqrt{16}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{16}

4

;

4^{2}

16

a_{2} = \frac{- 2 + 4}{2}

ⓘ The numerator and the denominator are computed before the division:

- 2 + 4

2

a_{2} = \frac{2}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{2}{2}

1

a_{2} = 1

Factors identified:

(a + 3) \cdot (a - 1)

ⓘ The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor:

\frac{a^{2} - 1}{a^{2} + 2 a - 3}

\frac{a + 1}{a + 3}

▶Simplified form:

\frac{a + 1}{a + 3}