4=3.095+(0.041x)+(-0.002x^2) — step-by-step solution

Let's solve in

ℝ

3.095 + 0.041 x - 0.002 x^{2} = 4

ⓘ We subtract

4

from both sides

3.095

+ 0.041 x

- 0.002 x^{2}

- 4 = 0

ⓘ Start from

3.095

and move back

4

on the number line, going below zero:

3.095 - 4

-0.905

- 0.905 + 0.041 x - 0.002 x^{2} = 0

- 0.002 x^{2} + 0.041 x - 0.905 = 0

Since

-1 \neq 0

, divide both sides by

-1

0.002 x^{2} - 0.041 x + 0.905 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 0.002

b = -0.041

and

c = 0.905

Δ = {(-0.041)}^{2} - 4 \cdot 0.002 \cdot 0.905

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

{(-0.041)}^{2}

(-0.041) \cdot (-0.041)

1.681 \times 10^{- 3}

■Multiplications are done before additions and subtractions:

4 \cdot 0.002

0.002

+ 0.002

+ 0.002

+ 0.002

0.008

Δ = 1.681 \times 10^{- 3} - 0.008 \cdot 0.905

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

0.008 \cdot 0.905

\frac{8 \cdot 905}{1,000,000}

\frac{7240}{1,000,000}

7.24 \times 10^{- 3}

Δ = 1.681 \times 10^{- 3} - 7.24 \times 10^{- 3}

ⓘ Start from

1.681 \times 10^{- 3}

and move back

7.24 \times 10^{- 3}

on the number line, going below zero:

1.681 \times 10^{- 3} - 7.24 \times 10^{- 3}

-5.559 \times 10^{- 3}

Δ = -5.559 \times 10^{- 3}

Δ < 0

, the equation has no real root.

▶Solution:

x \in \emptyset