Here we are given with a equation:
[tex] {x}^{3} = - 2197[/tex]
And we have to solve it with appropriate steps.
So, let's start solving.
Add 2197 to both sides of the equation,
[tex] {x}^{3} + 2197 = - 2197 + 2197[/tex]
[tex] {x}^{ 3} + 2197 = 0[/tex]
Now, using the identity:
- a³ + b³ = (a + b)(a² - ab + b²)
Let's proceed further.
x³ is the cube of x and 2197 is the cube of 13
[tex](x) {}^{3} + (13) {}^{3} = 0[/tex]
[tex](x + 13)( {x}^{2} - x \times 13 + {13}^{2} ) = 0[/tex]
[tex](x + 13)( {x}^{2} - 13x + 13) = 0[/tex]
Now,
- x + 13 = 0
- x² - 13x + 13 = 0
So, x = -13
And, for the second let's use the discriminate formula,
➝ D = b² - 4ac
➝ D = (-13)² - 4(1)(13)
➝ D = 169 - 52
➝ D = 117
Now, using the D. formula,
[tex]x = \dfrac{ - b \pm \sqrt{ D} }{2a} [/tex]
[tex]x = \dfrac{13 \pm \sqrt{117} }{2} [/tex]
[tex]x = \dfrac{13 \pm 3 \sqrt{13} }{2} [/tex]
So the roots of the equation are:
[tex]{ \boxed{ \bf{x = - 1 ,\: \frac{13 + 3 \sqrt{13} }{2} and \: \frac{13 - 3 \sqrt{13} }{2} }}}[/tex]
And we are done !!
#CarryOnLearning
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