Question

Solve this equation and show your work! (1 point for the correct answer and 1 point for showing how to solve the equation) x³ = -2,197

Answer 1

Here we are given with a equation:

${x}^{3} = - 2197$

And we have to solve it with appropriate steps.

So, let's start solving.

Add 2197 to both sides of the equation,

${x}^{3} + 2197 = - 2197 + 2197$

${x}^{ 3} + 2197 = 0$

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

$(x) {}^{3} + (13) {}^{3} = 0$

$(x + 13)( {x}^{2} - x \times 13 + {13}^{2} ) = 0$

$(x + 13)( {x}^{2} - 13x + 13) = 0$

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,

$x = \dfrac{ - b \pm \sqrt{ D} }{2a}$

$x = \dfrac{13 \pm \sqrt{117} }{2}$

$x = \dfrac{13 \pm 3 \sqrt{13} }{2}$

So the roots of the equation are:

${ \boxed{ \bf{x = - 1 ,\: \frac{13 + 3 \sqrt{13} }{2} and \: \frac{13 - 3 \sqrt{13} }{2} }}}$

And we are done !!

#CarryOnLearning

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Answer 2

Make the 3 into a fraction and then flip it to cancel out which makes it easier then do it to the -2,197 for you to get x