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

The original denominators must be nonzero:

m \in ℝ \ {- \frac{1}{2}, 1}

1) Let's simplify:

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

2) Let's factorize in

ℝ

m^{2} - 2 m + 1

You can rephrase the expression into:

{(m - 1)}^{2}

3) Let's factor in

ℝ

2 m^{2} - m - 1

Given the quadratic form

a m^{2} + b m + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = -1

and

c = -1

Δ = {(-1)}^{2} - 4 \cdot 2 \cdot (-1)

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

{(-1)}^{2}

(-1) \cdot (-1)

1

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

■Multiplying by -1 gives the opposite:

- 8 \cdot (-1)

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 2}

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

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

\sqrt{9}

3

;

3^{2}

9

ⅰ

2 \cdot 2

4

m_{1} = \frac{1 - 3}{4}

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

1 - 3

-2

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

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

2

, their greatest common divisor:

- \frac{2}{4}

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

- \frac{1}{2}

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

The quadratic formula gives

m_{2}

\frac{- b + \sqrt{Δ}}{2 a}

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

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

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

\sqrt{9}

3

;

3^{2}

9

ⅰ

2 \cdot 2

4

m_{2} = \frac{1 + 3}{4}

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

1 + 3

4

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

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

4

, their greatest common divisor:

\frac{4}{4}

1

m_{2} = 1

A trinomial with roots

m_{1}

and

m_{2}

a \cdot (m - m_{1}) \cdot (m - m_{2})

; bringing the coefficient into the brackets:

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

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

Factors identified:

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

We divide the numerator and denominator of

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

by the common factor:

m - 1

, which gives the new expression:

m - 1

▶Simplified form:

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