domain of y=(30-6x)/(x^2-11x+30)

We save the definition:

y = \frac{30 - 6 x}{x^{2} - 11 x + 30}

Let's find the domain of

\frac{30 - 6 x}{x^{2} - 11 x + 30}

For the expression

\frac{30 - 6 x}{x^{2} - 11 x + 30}

to be defined, we must have:

x^{2} - 11 x + 30 \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 = -11

and

c = 30

Δ = {(-11)}^{2} - 4 \cdot 1 \cdot 30

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

{(-11)}^{2}

(-11) \cdot (-11)

121

Δ = 121 - 4 \cdot 30

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 30

120

Δ = 121 - 120

ⓘ Start from

121

and move back

120

on the number line:

121 - 120

1

Δ = 1

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-11) - \sqrt{1}}{2 \cdot 1}

x_{1} = \frac{11 - 1}{2}

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

11 - 1

10

x_{1} = \frac{10}{2}

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

2

, their greatest common divisor:

\frac{10}{2}

5

x_{1} = 5

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-11) + \sqrt{1}}{2 \cdot 1}

x_{2} = \frac{11 + 1}{2}

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

11 + 1

12

x_{2} = \frac{12}{2}

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

2

, their greatest common divisor:

\frac{12}{2}

6

x_{2} = 6

▶Domain of definition:

x \in ℝ \ {5, 6}