x^2+12x^2+48x+64=0 — step-by-step solution

Let's solve in

ℝ

x^{2}

+ 12 x^{2}

+ 48 x

+ 64 = 0

ⓘ Like terms are grouped by adding their coefficients:

x^{2} + 12 x^{2}

13 x^{2}

13 x^{2} + 48 x + 64 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 13

b = 48

and

c = 64

Δ = 48^{2} - 4 \cdot 13 \cdot 64

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

48^{2}

48 \cdot 48

2304

■Multiplications are done before additions and subtractions:

4 \cdot 13

4 \cdot (10 + 3)

4 \cdot 10 + 4 \cdot 3

40 + 12

52

Δ = 2304 - 52 \cdot 64

ⓘ Multiplications are done before additions and subtractions:

52 \cdot 64

(50 + 2) \cdot 64

50 \cdot 64 + 2 \cdot 64

3200 + 128

3328

Δ = 2304 - 3328

ⓘ Start from

2304

and move back

3328

on the number line, going below zero:

2304 - 3328

-1024

Δ = -1024

Δ < 0

, the equation has no real root.

▶Solution:

x \in \emptyset