3x^2+249=-54x — step-by-step solution

Let's solve in

ℝ

3 x^{2} + 249 = -54 x

Since

3 \neq 0

, divide both sides by

3

x^{2} + 83 = - 18 x

x^{2} + 18 x + 83 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 18

and

c = 83

Δ = 18^{2} - 4 \cdot 1 \cdot 83

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

18^{2}

18 \cdot 18

324

Δ = 324 - 4 \cdot 83

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 83

4 \cdot (80 + 3)

4 \cdot 80 + 4 \cdot 3

320 + 12

332

Δ = 324 - 332

ⓘ Start from

324

and move back

332

on the number line, going below zero:

324 - 332

-8

Δ = -8

Δ < 0

, the equation has no real root.

▶Solution:

x \in \emptyset