Best solutions to - "Quadratic equation 3x3=24"

Advadlippabrj

Answered question

2022-01-23

Quadratic equation

3 x^{3} = 24

fionaluvsyou0x

Beginner2022-01-24Added 11 answers

Step 1
May be you know this already but its

oferenteoo

Beginner2022-01-25Added 12 answers

Step 1
Hints

3 x^{3} = 24 \Leftrightarrow x^{3} - 8 = 0

\frac{x^{3} - 8}{x - 2} = x^{2} + 2 x + 4

Added: The second equation means that

x^{3} - 8 = (x - 2) (x^{2} + 2 x + 4)

Thus we have

x^{3} - 8 = 0 \Leftrightarrow (x - 2) (x^{3} + 2 x + 4) = 0

whose solutions are the values of x that make the factor

x - 2

or the factor

x^{2} + 2 x + 4

equal to 0. The equation

x - 2 = 0

has the solution

x = 2

. The solutions of the equation

x^{2} + 2 x + 4 = 0

may be found by the quadratic formula or as follows:

x^{2} + 2 x + 4 = 0 \Leftrightarrow {(x + 1)}^{2} + 3 = 0 \Leftrightarrow {(x + 1)}^{2} = - 3

\Leftrightarrow x + 1 = \pm \sqrt{- 3} = \pm \sqrt{3} i

\Leftrightarrow x = - 1 \pm \sqrt{3} i

x^{3} - 8 = 0 \Leftrightarrow x = 2

x = - 1 \pm \sqrt{3} i

RizerMix

Expert2022-01-27Added 656 answers

factor using polynomial division:

\frac{x^{3} - 8}{x - 2} = (x^{2} + 2 x + 4)

To solve this, we can quickly complete the square as follows: Notice that we are setting

x^{2} + 2 x + 4 = 0

(x + 1)^{2} + 3 = 0

(x + 1)^{2} = - 3

x + 1 = \pm i \sqrt{3}

x = - 1 \pm i \sqrt{3}

and we have our answer. I find this slightly faster than using the quadratic formula.

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