0.2*y*(1-(y/5))-0.24=0 — step-by-step solution

Let's solve in

ℝ

0.2 y \cdot (1 - \frac{y}{5}) - 0.24 = 0

ⓘ Expand by multiplying each term inside the parentheses:

0.2 y \cdot (1 - \frac{y}{5})

0.2 y + 0.2 y \cdot (- \frac{y}{5})

0.2 y + 0.2 y \cdot (- \frac{y}{5}) - 0.24 = 0

ⓘ Multiplying a fraction by a term multiplies its numerator:

0.2 y \cdot (- \frac{y}{5})

- \frac{0.2 \cdot y \cdot y}{5}

- \frac{0.2 y^{2}}{5}

0.2 y - \frac{0.2 y^{2}}{5} - 0.24 = 0

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

- \frac{0.2 y^{2}}{5}

- 0.04 y^{2}

0.2 y - 0.04 y^{2} - 0.24 = 0

- 0.04 y^{2} + 0.2 y - 0.24 = 0

Since

-1 \neq 0

, divide both sides by

-1

0.04 y^{2} - 0.2 y + 0.24 = 0

Given the quadratic form

a y^{2} + b y + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 0.04

b = -0.2

and

c = 0.24

Δ = {(-0.2)}^{2} - 4 \cdot 0.04 \cdot 0.24

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

{(-0.2)}^{2}

(-0.2) \cdot (-0.2)

0.04

■Multiplications are done before additions and subtractions:

4 \cdot 0.04

0.04

+ 0.04

+ 0.04

+ 0.04

0.16

Δ = 0.04 - 0.16 \cdot 0.24

ⓘ Multiplications are done before additions and subtractions. Multiply without the decimal points, then put the point back by counting the decimals of both factors:

0.16 \cdot 0.24

\frac{16 \cdot 24}{10,000}

\frac{384}{10,000}

0.0384

Δ = 0.04 - 0.0384

ⓘ Start from

0.04

and move back

0.0384

on the number line:

0.04 - 0.0384

0.0016

Δ = 0.0016

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

y_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-0.2) - \sqrt{0.0016}}{2 \cdot 0.04}

y_{1} = \frac{0.2 - \sqrt{0.0016}}{2 \cdot 0.04}

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

\sqrt{0.0016}

0.04

;

{0.04}^{2}

0.0016

ⅰ

2 \cdot 0.04

0.04 + 0.04

0.08

y_{1} = \frac{0.2 - 0.04}{0.08}

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

0.2 - 0.04

0.16

y_{1} = \frac{0.16}{0.08}

ⓘ The numerator and the denominator are multiplied by

100

to remove the decimal point, then divided by

8

, their greatest common divisor:

\frac{0.16}{0.08}

\frac{0.16 \cdot 100}{0.08 \cdot 100}

\frac{16}{8}

2

y_{1} = 2