25x^2-95x-20=0 — step-by-step solution

1) Let's solve in

ℝ

25 x^{2} - 95 x - 20 = 0

Since

5 \neq 0

, divide both sides by

5

5 x^{2} - 19 x - 4 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 5

b = -19

and

c = -4

Δ = {(-19)}^{2} - 4 \cdot 5 \cdot (-4)

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

{(-19)}^{2}

(-19) \cdot (-19)

361

■Multiplications are done before additions and subtractions:

4 \cdot 5

20

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

- 20 \cdot (-4)

20 \cdot 4

Δ = 361 + 20 \cdot 4

ⓘ Multiplications are done before additions and subtractions:

20 \cdot 4

80

Δ = 361 + 80

ⓘ Start from

361

and move forward

80

on the number line:

361 + 80

441

Δ = 441

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-19) - \sqrt{441}}{2 \cdot 5}

x_{1} = \frac{19 - \sqrt{441}}{2 \cdot 5}

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

\sqrt{441}

21

;

21^{2}

441

ⅰ

2 \cdot 5

10

x_{1} = \frac{19 - 21}{10}

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

19 - 21

-2

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

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

2

, their greatest common divisor:

- \frac{2}{10}

- \frac{1 \cdot 2}{5 \cdot 2}

- \frac{1}{5}

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-19) + \sqrt{441}}{2 \cdot 5}

x_{2} = \frac{19 + \sqrt{441}}{2 \cdot 5}

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

\sqrt{441}

21

;

21^{2}

441

ⅰ

2 \cdot 5

10

x_{2} = \frac{19 + 21}{10}

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

19 + 21

40

x_{2} = \frac{40}{10}

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

10

, their greatest common divisor:

\frac{40}{10}

4

x_{2} = 4

▶Solutions obtained:

x = {- \frac{1}{5}, 4}

2) The numeric value of:

x = {- \frac{1}{5}, 4}

is:

▷

x = {-0.2, 4}