x^2-3x+2<= 0 — step-by-step solution

Let's solve in

ℝ

x^{2} - 3 x + 2 \leq 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -3

and

c = 2

Δ = {(-3)}^{2} - 4 \cdot 1 \cdot 2

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

{(-3)}^{2}

(-3) \cdot (-3)

9

Δ = 9 - 4 \cdot 2

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 2

8

Δ = 9 - 8

ⓘ Start from

9

and move back

8

on the number line:

9 - 8

1

Δ = 1

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-3) - \sqrt{1}}{2 \cdot 1}

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

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

3 - 1

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-3) + \sqrt{1}}{2 \cdot 1}

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

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

3 + 1

4

x_{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

x_{2} = 2

The quadratic has the opposite sign to a =

1

between its roots, so it is indeed negative between its roots; the roots, where it is zero, are included.

▶Solution obtained:

x \in [1, 2]