intercepts of f(x)=(2x^2-3x-20)/(x-5)

On the horizontal axis, y = 0: solve

\frac{2 x^{2} - 3 x - 20}{x - 5} = 0

1) Let's solve in

ℝ

\frac{2 x^{2} - 3 x - 20}{x - 5} = 0

A quotient is zero if and only if its numerator is zero and its denominator is not: we solve the numerator equal to 0

⇥1.1) Let's solve in

ℝ

2 x^{2} - 3 x - 20 = 0

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = -3

and

c = -20

⇥

Δ = {(-3)}^{2} - 4 \cdot 2 \cdot (-20)

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

{(-3)}^{2}

(-3) \cdot (-3)

9

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

■Two minus signs cancel out, the product of two negative numbers is positive:

- 8 \cdot (-20)

8 \cdot 20

⇥

Δ = 9 + 8 \cdot 20

⇥ⓘ Multiplications are done before additions and subtractions:

8 \cdot 20

160

⇥

Δ = 9 + 160

⇥ⓘ Start from

9

and move forward

160

on the number line:

9 + 160

169

⇥

Δ = 169

⇥As

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-3) - \sqrt{169}}{2 \cdot 2}

⇥

x_{1} = \frac{3 - \sqrt{169}}{2 \cdot 2}

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

\sqrt{169}

13

;

13^{2}

169

⇥ⅰ

2 \cdot 2

4

⇥

x_{1} = \frac{3 - 13}{4}

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

3 - 13

-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{- (-3) + \sqrt{169}}{2 \cdot 2}

⇥

x_{2} = \frac{3 + \sqrt{169}}{2 \cdot 2}

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

\sqrt{169}

13

;

13^{2}

169

⇥ⅰ

2 \cdot 2

4

⇥

x_{2} = \frac{3 + 13}{4}

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

3 + 13

16

⇥

x_{2} = \frac{16}{4}

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

4

, their greatest common divisor:

\frac{16}{4}

4

⇥

x_{2} = 4

⇥Candidates, pending the domain check:

x = {- \frac{5}{2}, 4}

⇥1.2) To avoid a zero denominator, let's solve:

x - 5 \neq 0

⇥Domain condition:

x \neq 5

For the expression

\frac{2 x^{2} - 3 x - 20}{x - 5}

to be defined, we must have:

x - 5 \neq 0

Domain constraints:

x \neq 5

The graph crosses the horizontal axis at

{(- \frac{5}{2}, 0), (4, 0)}

On the vertical axis, x = 0: calculate f(0)

⇥

\frac{2 \cdot 0^{2} - 3 \cdot 0 - 20}{0 - 5}

⇥ⓘ A product with a zero factor is zero:

2 \cdot 0

0

-3 \cdot 0

0

■Adding 0 changes nothing:

0 - 5

-5

■The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{0 + 0 - 20}{-5}

- \frac{0 + 0 - 20}{5}

⇥

- \frac{0 + 0 - 20}{5}

⇥ⓘ Adding 0 changes nothing:

0 + 0 - 20

-20

■Two minus signs cancel out:

- (\frac{-20}{5})

\frac{20}{5}

⇥

\frac{20}{5}

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

5

, their greatest common divisor:

\frac{20}{5}

4

⇥

4

▶Axis intercepts:

{(- \frac{5}{2}, 0), (4, 0), (0, 4)}