solve x^2-5x+6=0 — step-by-step solution

Let's solve in

ℝ

x^{2} - 5 x + 6 = 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 = -5

and

c = 6

Δ = {(-5)}^{2} - 4 \cdot 1 \cdot 6

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

{(-5)}^{2}

(-5) \cdot (-5)

25

Δ = 25 - 4 \cdot 6

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 6

24

Δ = 25 - 24

ⓘ Start from

25

and move back

24

on the number line:

25 - 24

1

Δ = 1

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-5) - \sqrt{1}}{2 \cdot 1}

x_{1} = \frac{5 - 1}{2}

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

5 - 1

4

x_{1} = \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_{1} = 2

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-5) + \sqrt{1}}{2 \cdot 1}

x_{2} = \frac{5 + 1}{2}

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

5 + 1

6

x_{2} = \frac{6}{2}

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

2

, their greatest common divisor:

\frac{6}{2}

3

x_{2} = 3

▶Solutions obtained:

x = {2, 3}