simplify (m^2)/(m^2-4)+1/(2-m) — step-by-step solution

The original denominators must be nonzero:

m \in ℝ \ {-2, 2}

1) Let's add fractions:

\frac{m^{2}}{m^{2} - 4} + \frac{1}{2 - m}

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{m^{2}}{m^{2} - 4} + \frac{1}{2 - m}

\frac{m^{2} \cdot (2 - m) + (m^{2} - 4)}{(m^{2} - 4) \cdot (2 - m)}

\frac{m^{2} \cdot (2 - m) + (m^{2} - 4)}{(m^{2} - 4) \cdot (2 - m)}

ⓘ What the terms have in common is factored out, so that the fraction can be simplified:

\frac{m^{2} \cdot (2 - m) + m^{2} - 4}{(m^{2} - 4) \cdot (2 - m)}

\frac{(- m + 2) \cdot (m^{2} - 1 \cdot (2 + m))}{(m^{2} - 4) \cdot (2 - m)}

\frac{(- m + 2) \cdot (m^{2} - 1 \cdot (2 + m))}{(m^{2} - 4) \cdot (2 - m)}

With the formula:

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

, we divide the numerator and denominator of

\frac{- m + 2}{m^{2} - 4}

by the common factor:

- m + 2

, which gives the new fraction:

\frac{1}{- m - 2}

\frac{m^{2} - 1 \cdot (2 + m)}{(- m - 2) \cdot (2 - m)}

ⓘ Multiplying by -1 gives the opposite:

-1 \cdot (2 + m)

- (2 + m)

■The minus sign is factored out, changing the sign of each term:

- m - 2

- (m + 2)

\frac{m^{2} - (2 + m)}{- (m + 2) \cdot (2 - m)}

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

- (2 + m)

- 2 - m

\frac{m^{2} - 2 - m}{- (m + 2) \cdot (2 - m)}

ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{m^{2} - 2 - m}{- (m + 2) \cdot (2 - m)}

- \frac{m^{2} - 2 - m}{(m + 2) \cdot (2 - m)}

- \frac{m^{2} - 2 - m}{(m + 2) \cdot (2 - m)}

2) Let's factorize in

ℝ

m^{2} - 2 - m

Given the quadratic form

a m^{2} + b m + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -1

and

c = -2

Δ = {(-1)}^{2} - 4 \cdot 1 \cdot (-2)

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

{(-1)}^{2}

(-1) \cdot (-1)

1

Δ = 1 - 4 \cdot (-2)

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

- 4 \cdot (-2)

8

Δ = 1 + 8

ⓘ Start from

1

and move forward

8

on the number line:

1 + 8

9

Δ = 9

Δ > 0

, the equation has two roots:

The quadratic formula gives

m_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-1) - \sqrt{9}}{2 \cdot 1}

m_{1} = \frac{1 - \sqrt{9}}{2}

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

\sqrt{9}

3

;

3^{2}

9

m_{1} = \frac{1 - 3}{2}

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

1 - 3

-2

m_{1} = - \frac{2}{2}

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

2

, their greatest common divisor:

- \frac{2}{2}

-1

m_{1} = -1

The quadratic formula gives

m_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-1) + \sqrt{9}}{2 \cdot 1}

m_{2} = \frac{1 + \sqrt{9}}{2}

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

\sqrt{9}

3

;

3^{2}

9

m_{2} = \frac{1 + 3}{2}

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

1 + 3

4

m_{2} = \frac{4}{2}

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

2

, their greatest common divisor:

\frac{4}{2}

2

m_{2} = 2

Factors found:

(m + 1) \cdot (m - 2)

We simplify the fraction:

\frac{m - 2}{2 - m}

by the common factor:

m - 2

to get the new expression:

-1

- \frac{m + 1}{(m + 2) \cdot (-1)}

ⓘ Multiplying by -1 gives the opposite:

(m + 2) \cdot (-1)

- (m + 2)

- \frac{m + 1}{- (m + 2)}

ⓘ Two minus signs cancel out:

- \frac{m + 1}{- (m + 2)}

\frac{m + 1}{m + 2}

▶Result:

\frac{m + 1}{m + 2}