Introduction
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In many mathematical analyses, we consider a simple quadratic equation:
[latex]ax^2 + bx + c = 0[/latex]
where a, b, and c are constants.
Section 1: Algebraic Identities
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Basic algebraic identities include:
[latex](a + b)^2 = a^2 + 2ab + b^2[/latex]
[latex](a – b)^2 = a^2 – 2ab + b^2[/latex]
[latex](a + b)(a – b) = a^2 – b^2[/latex]
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Section 2: Calculus Overview
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The derivative of a function is defined as:
[latex]\frac{df}{dx} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) – f(x)}{\Delta x}[/latex]
Similarly, the integral of a function is represented by:
[latex]\int_a^b f(x),dx = F(b) – F(a)[/latex]
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Section 3: Geometry and Trigonometry
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For a right triangle with hypotenuse c and legs a and b:
[latex]c^2 = a^2 + b^2[/latex]
Trigonometric identities include:
[latex]\sin^2\theta + \cos^2\theta = 1[/latex]
[latex]\tan\theta = \frac{\sin\theta}{\cos\theta}[/latex]
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Section 4: Probability and Statistics
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The mean of a dataset is:
[latex]\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i[/latex]
The variance is given by:
[latex]\sigma^2 = \frac{1}{n}\sum_{i=1}^{n} (x_i – \bar{x})^2[/latex]
And the standard deviation is simply:
[latex]\sigma = \sqrt{\sigma^2}[/latex]
Section 5: Physics Equations
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Newton’s second law of motion:
[latex]F = ma[/latex]
Einstein’s famous energy equation:
[latex]E = mc^2[/latex]
Gravitational force between two masses:
[latex]F = G\frac{m_1 m_2}{r^2}[/latex]
Electromagnetic wave equation:
[latex]\nabla^2 E – \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2} = 0[/latex]
Section 6: Advanced Mathematics
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Fourier transform of a signal:
[latex]\mathcal{F}{f(t)} = \int_{-\infty}^{\infty} f(t)e^{-i 2 \pi f t},dt[/latex]
Laplace transform:
[latex]\mathcal{L}{f(t)} = \int_0^{\infty} e^{-st}f(t),dt[/latex]
Determinant of a 2×2 matrix:
[latex]\det\begin{pmatrix}a & b \ c & d\end{pmatrix} = ad – bc[/latex]
Section 7: Miscellaneous Examples
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Binomial theorem:
[latex](x + y)^n = \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k[/latex]
Euler’s formula:
[latex]e^{i\pi} + 1 = 0[/latex]
Taylor series expansion:
[latex]f(x) = f(a) + f'(a)(x – a) + \frac{f”(a)}{2!}(x – a)^2 + \cdots[/latex]
Conclusion
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Mathematics, physics, and engineering all use formulas like:
[latex]\sum_{i=1}^{n} i = \frac{n(n+1)}{2}[/latex]
to represent the sum of the first n natural numbers.
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