Cubic Formula Proof Step 4: Other Solutions of y using the Cube Root of Unity

in #hive-128780 • 3 days ago

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In this video I go over the fourth step of the Cubic Formula proof which involves obtaining the complex solutions of y by considering the cube root of unity. Since the cube root of the z values from Vieta's substitution has 3 factors, we need to multiply the principle solution by the factors to obtain each separate solution. We solved the first solution already in the previous video, and that solution is termed the principle solution since it corresponds to the factor 1. The next two factors are complex numbers, but when we multiply them by z, we can apply the inverse relationship to simplify the equations. The result is just the principle solution of y with each term multiplied by one complex factor; the order is flipped for the other solution.

In the next video, I will plug these y-values back into the PQ substitution to obtain our x solutions.

6 Step 4 Find other Solutions of y.jpeg

Timestamps

  • Step 4: Find the other solutions of y using the cube root of unity: 0:00
  • First solution is just our previous or principle solution of y: 2:00
  • Next two solutions involve multiplying by the complex roots: 2:36
  • Solving the second solution of y: 3:26
  • Solving the third solution of y: 6:28

Notes and playlists

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    !summarize

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      Part 5/5:

      As we continue to explore mathematical concepts, the techniques discussed here empower us to tackle increasingly sophisticated problems, using fundamental principles of unity, complex numbers, and cubic functions.

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        Part 1/5:

        Finding Multiple Solutions with Cube Roots of Unity

        In our journey into the world of complex numbers and cube roots, we have reached a pivotal stage where we explore the application of cube roots of unity to discover multiple solutions for the equation involving an unknown variable ( y ).

        Understanding Cube Roots of Unity

        To start, cube roots of unity are solutions to the equation ( x^3 = 1 ). These roots are crucial when working with cubic equations and allow us to break down complex problems into simpler components. In essence, they give us three distinct solutions: 1 (the principal root), and two other complex roots, typically denoted as ( W_1 ) and ( W_2 ).

        The Primary Solution

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          Part 2/5:

          We begin by considering one of the cube roots ( z ), expressed as ( \sqrt[3]{Z} ). By applying the cube roots of unity, we can generate three solutions for the variable ( z ):

          1. The principal cube root, denoted as ( z_1 ), which is equal to ( \sqrt[3]{Z} ).

          2. The second solution, ( z_2 ), is represented as ( z_1 \times W_1 ), where ( W_1 ) is one of the cube roots of unity.

          3. Finally, the third solution, ( z_3 ), is derived as ( z_1 \times W_2 ), introducing the last cube root of unity into our calculations.

          Converting ( z ) Solutions to ( y ) Solutions

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            Part 4/5:

            Similarly, we compute ( y_3 ):

            [

            y_3 = \frac{z_3 - p}{3z_3}

            ]

            Replacing ( z_3 ) with ( z_1 \times W_2 ):

            [

            y_3 = \frac{(z_1 \times W_2) - p}{3(z_1 \times W_2)}

            ]

            This equation mirrors the complexity of ( y_2 ), demonstrating how the intricate relationships between cube roots translate back into the quotient required for ( y ).

            Conclusion

            Through the methodical breakdown of cube roots and their application to complex equations, we have successfully modeled the relationships between ( y ) and its corresponding ( z ) values guided by cube roots of unity. Each derived solution illustrates the elegance of algebraic manipulation and the beauty inherent in solving cubic equations.

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