domain of f(x)=(3x-4)/(x^2-5x+10)

We save the definition:

f (x) = \frac{3 x - 4}{x^{2} - 5 x + 10}

Let's find the domain of

\frac{3 x - 4}{x^{2} - 5 x + 10}

For the expression

\frac{3 x - 4}{x^{2} - 5 x + 10}

to be defined, we must have:

x^{2} - 5 x + 10 \neq 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 = 10

Δ = {(-5)}^{2} - 4 \cdot 1 \cdot 10

ⓘ 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 10

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 10

40

Δ = 25 - 40

ⓘ Start from

25

and move back

40

on the number line, going below zero:

25 - 40

-15

Δ = -15

Δ < 0

, the equation has no real root.

▶Domain of definition:

x \in ℝ