domain of f(x)=(3x^2-3)/(2x^2+7x+5)

We save the definition:

f (x) = \frac{3 x^{2} - 3}{2 x^{2} + 7 x + 5}

1) Let's find the domain of

\frac{3 x^{2} - 3}{2 x^{2} + 7 x + 5}

For the expression

\frac{3 x^{2} - 3}{2 x^{2} + 7 x + 5}

to exist, you must check:

2 x^{2} + 7 x + 5 \neq 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = 7

and

c = 5

Δ = 7^{2} - 4 \cdot 2 \cdot 5

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

7^{2}

7 \cdot 7

49

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

Δ = 49 - 8 \cdot 5

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 5

40

Δ = 49 - 40

ⓘ Start from

49

and move back

40

on the number line:

49 - 40

9

Δ = 9

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 7 - \sqrt{9}}{2 \cdot 2}

x_{1} = \frac{- 7 - \sqrt{9}}{2 \cdot 2}

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

\sqrt{9}

3

;

3^{2}

9

ⅰ

2 \cdot 2

4

x_{1} = \frac{- 7 - 3}{4}

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

- 7 - 3

-10

x_{1} = - \frac{10}{4}

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

2

, their greatest common divisor:

- \frac{10}{4}

- \frac{5 \cdot 2}{2 \cdot 2}

- \frac{5}{2}

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 7 + \sqrt{9}}{2 \cdot 2}

x_{2} = \frac{- 7 + \sqrt{9}}{2 \cdot 2}

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

\sqrt{9}

3

;

3^{2}

9

ⅰ

2 \cdot 2

4

x_{2} = \frac{- 7 + 3}{4}

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

- 7 + 3

-4

x_{2} = - \frac{4}{4}

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

4

, their greatest common divisor:

- \frac{4}{4}

-1

x_{2} = -1

▶Domain of definition:

x \in ℝ \ {- \frac{5}{2}, -1}

2) The numeric value of:

x \in ℝ \ {- \frac{5}{2}, -1}

is:

▷

x \in ℝ \ {-2.5, -1}