Video: Solving Cubic Equations over the Set of Real Numbers

Solve the following equation (54/π‘₯Β³) + 1,814 = βˆ’186, given that π‘₯ is in ℝ.

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Video Transcript

Solve the following equation: 54 over π‘₯ cubed plus 1814 is equal to negative 186, given that π‘₯ is a real number.

We can begin this question by subtracting 1814 from both sides of the equation. This gives us 54 over π‘₯ cubed is equal to negative 186 minus 1814. The right-hand side simplifies to negative 2000. We can then multiply both sides of this equation by π‘₯ cubed such that 54 is equal to negative 2000π‘₯ cubed. Dividing both sides by negative 2000 gives us negative 54 over 2000, which simplifies to negative 27 over 1000.

As cube rooting is the opposite of cubing, our final step is to cube root both sides of this equation. We recall that the cube root of any negative number gives a negative answer. The cube root of negative π‘₯ is equal to the negative of the cube root of π‘₯. When cube rooting any fraction, we can also cube root the numerator and denominator separately.

Using these two rules, π‘₯ is equal to the negative of the cube root of 27 divided by the cube root of 1000. The cube root of 27 is equal to three as three cubed is 27. The cube root of 1000 is 10 as 10 cubed is 1000. π‘₯ is therefore equal to negative three-tenths or negative 0.3.

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