-x^2-x+2>= 0 — step-by-step solution

Let's solve in

ℝ

- x^{2} - x + 2 \geq 0

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 = -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

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

-4 \cdot (-1)

4

Δ = 1 + 4 \cdot 2

ⓘ Multiplications are done before additions and subtractions:

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

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

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

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

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

\sqrt{9}

3

;

3^{2}

9

■A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

2 \cdot (-1)

-2

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

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

\frac{1 - 3}{-2}

- \frac{1 - 3}{2}

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

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

1 - 3

-2

x_{1} = - (\frac{-2}{2})

ⓘ Two minus signs cancel out:

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

\frac{2}{2}

x_{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

x_{1} = 1