Section 1.1: Background Information

In elementary algebra, students learn the equation of a parabola, represented as a second-order polynomial. The general solution that equates this polynomial to zero (0) is encapsulated in the quadratic formula, as expressed in the following equation:

A noteworthy aspect of the quadratic formula is the discriminant, defined as:

The discriminant is crucial in determining the number of potential solutions based on specific criteria outlined in:

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• • Discriminant Value

    • Number of Solutions

    • • Less than 0
      • No real solutions
    • • Equal to 0
      • One real solution
    • • Greater than 0
      • Two distinct real solutions

These criteria can often be understood intuitively; for instance, a negative discriminant indicates the absence of real solutions. Once the discriminant is computed, taking its square root reveals whether the outcome is a real number or a complex number.

With this foundational knowledge established, let's explore some case studies drawn from the Algebra and Calculus textbook.

Section 1.2: Case Studies

As in my previous writings, I will utilize the JupyterLab desktop application to generate graphs (Bektas 2021). Prior to this, I will make some adjustments to my notebook concerning the function used for plotting:

def plot_functions(f_l, f_r, r=[0, 100, 0.1]):

plt.figure()

# [... snip ...]

plt.plot(np.arange(r[0], r[1], r[2]),

[f_l(x) for x in np.arange(r[0], r[1], r[2])], color="black", label=r'$f_L(x)$')

plt.plot(np.arange(r[0], r[1], r[2]),

[f_r(x) for x in np.arange(r[0], r[1], r[2])], "--", color="blue", label=r'$f_R(x)$')

# [... snip ...]

I modified the function parameters to streamline the bounds for both functions, simplifying the plotting process.

Let's examine the first problem:

Q10: "(x+2)(x-3) = x²-x-6"

Before solving this, I will visualize it on a Cartesian graph to determine its proof-worthiness. I start by breaking down the left and right sides of the equation into two distinct functions:

Subsequently, I implement these functions in Python:

f_L = lambda x: (x + 2) * (x - 3)

f_R = lambda x: x**2 - x - 6

plot_functions(f_L, f_R, [-100, 100, 0.01])

The output of the graph is illustrated in Figure 1.

This equality appears to be a strong candidate for proof. I will apply the quadratic formula to find the roots of the right side of the equation, as shown in Figure 2.

The roots derived from the quadratic formula yield curious coefficients. If these coefficients are rational or real, I must question the accuracy of my calculations. To verify, I will apply the distributive property to the left side of the equation, as represented in Figure 3.

It’s possible that an error occurred in my application of the quadratic formula, similar to my findings in Q10. This serves as a reminder to utilize multiple strategies to double-check our results.

Q11: "(x-3)(x+3) = x²+9"

Next, I will break this equality down into two functions, just as before:

I will encode these functions for graphing purposes:

f_L = lambda x: (x - 3) * (x + 3)

f_R = lambda x: x**2 + 9

plot_functions(f_L, f_R, [-100, 100, 0.01])

The output is captured in Figure 4.

Similar to the previous case study, this equality seems promising. I will attempt to prove it via the quadratic method, and if the results are unexpected, I will also apply the distributive law to the factored side, as shown in Figure 5.

For this analysis, I will avoid complex numbers, opting instead to use the distributive law on the coefficients from the factored form to validate the equality with the quadratic.

Q4 (again): "(x + 1)(x + 3) = x² + 4x + 3"

Previously, I disproved this equality in another article ("Aleksey" 2022b), but I will seize this opportunity to apply the quadratic formula:

The results from the quadratic formula are shown in Figure 7.

Although the derived coefficients from this formula do not match those of the left side, I must ensure that I applied the formula correctly, as I did in the earlier example.

End Matter

The quadratic formula may seem intricate and sometimes leads to non-real complex solutions. Readers might wonder, "Why should one utilize it?"

In my view, one key reason is that individuals tackling such problems may not have the factored form readily available. The quadratic formula can be particularly useful when traditional factoring methods fail. Additionally, it serves as a valuable tool for cross-verifying results—human errors in mathematics are not uncommon, making it prudent to double-check one's calculations.

Throughout this article, I encountered numerous mistakes related to the quadratic formula. However, they exemplify the importance of employing various methods for verification. Interested readers can access the notebook used for generating these graphs here: [Plug].

Furthermore, if you're engaged in textbook mathematics, computational thinking is an essential mindset. For any readers who enjoy watching problem-solving in action, I encourage you to explore my series of technical articles where I tackle computer science and security engineering challenges.

References

Amiran, E. Y. (2014). Algebra and Calculus: Mathematical Modeling for Business, Economics, and Finance. CreateSpace Independent Publishing Platform.

In this video titled Proof of the Quadratic Formula - Easy Explanation, the presenter offers a clear and engaging explanation of the quadratic formula and its derivation.

Another insightful video, Quadratic formula (proof) | Quadratic equations | Algebra I | Khan Academy, provides a thorough walkthrough of the quadratic formula and its applications.