Quadratic Formula by the PQ Substitution Method

mes in #hive-128780 • 6 days ago

In this video I solve the quadratic formula again, but this time using the PQ method instead of the complete the squares method that I used earlier. This is because the PQ method is utilized throughout the cubic formula proof, so this is a good preview. The procedure starts off the same as before, by dividing out the quadratic equation by a, but now we set p = b/a and q = c/a. We can now shift the parabola, which is just a quadratic equation, by setting x = y + k in order to cancel out the px term. Expanding out, we find that the value k = -p/2 cancels out the px term, thus allowing us to solve for y. Plugging y back into x, we obtain the PQ version of the quadratic formula.

1 Quadratic Formula PQ Method.jpeg

Time stamps

Solve quadratic formula using PQ substitution: 0:00
Set p = b/a and q = c/a to get PQ version of quadratic equation: 0:40
Shift parabola to remove px term by setting x = y + k: 1:02
Set k = -p/2 to cancel out px terms: 2:33
Rearrange to solve for y: 5:13
Plug y back into x to obtain PQ formula: 5:37
PQ formula: 6:30

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mes 6 days ago

Summary: https://peakd.com/hive-128780/@mes/oqzqzhqo
Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EF05ExjzLbB64NgUjoV1hl
Notes: https://peakd.com/hive-128780/@mes/dzekfnxh
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mes 6 days ago

!summarize

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ai-summaries 6 days ago

Part 4/4:

results in the final expression for ( x):

[ x = \frac{p}{2} \pm \sqrt{\frac{p^2}{4} - q} ]

Conclusion: Validating the PQ Method

The PQ substitution method is encapsulated effectively in the final boxed expression, providing both a visual and conceptual means of solving quadratic equations. Additionally, this method serves as a foundational step in the pursuit of deriving the cubic formula, demonstrating its essential role in higher-order polynomial solutions.

As we continue to explore the realms of algebra, methods like the PQ substitution strengthen our understanding and problem-solving capabilities, paving the way for tackling more complex equations with confidence.

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ai-summaries 6 days ago

Part 1/4:

Understanding the PQ Substitution Method for Quadratic Equations

The PQ substitution method presents an innovative way of solving quadratic equations, offering a fresh perspective that aids in the derivation of the cubic formula.

Step Zero: Introducing the PQ Method

The initial step—referred to as Step Zero—involves a systematic approach to handling quadratic equations expressed in the standard form:

[ ax^2 + bx + c = 0 ]

To utilize the PQ substitution method, the first task is to normalize the equation. This entails dividing every term by ( a ), the coefficient of the quadratic term. This results in the transformed equation:

[ x^2 + px + q = 0 ]

where ( p = \frac{b}{a} ) and ( q = \frac{c}{a} ).

Shifting the Parabola

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ai-summaries 6 days ago

Part 2/4:

The essence of the PQ method lies in shifting the parabola defined by the quadratic equation. The goal of this shift is to eliminate the linear term ( px ), simplifying the expression further.

To achieve this, a new variable ( y ) is introduced through the substitution:

[ x = y + k ]

where ( k ) is a constant chosen to eliminate the ( px ) term. Upon substituting ( x ) with ( y + k ) in our equation, we engage in expanding the terms to reveal a new form of the quadratic equation.

Establishing the Value of K

Through the expansion process, we notice key terms that emerge:

[ (y + k)^2 + p(y + k) + q = 0 ]

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