Quadratic Equation, Binomial Theorem & Logarithms

Quadratic Equation, Quartic Equation, Binomial Theorem, and Logarithms

Quadratic equation

A quadratic equation is a second-degree polynomial equation. Its general form is \( Ax^2 + Bx + C = 0 \), where \( A \), \( B \), and \( C \) are real numbers and \( A \neq 0 \).

Pure quadratic equation occurs when the middle coefficient is zero (\( B = 0 \)). In this case, the two roots are numerically equal but have opposite signs (\( r_1 = -r_2 \)).

The discriminant is the value \( B^2 - 4AC \) found inside the radical of the quadratic formula. It strictly determines the nature of the roots:

• When \( B^2 - 4AC = 0 \), the roots are real and equal (there is exactly one distinct root).
• When \( B^2 - 4AC > 0 \), the roots are real and unequal.
• When \( B^2 - 4AC < 0 \), the roots are imaginary and unequal.

There are reliable properties relating the roots \( r_1 \) and \( r_2 \) to the coefficients of the equation. The sum of the two roots is always \( \dfrac{-B}{A} \), and the product of the two roots is always \( \dfrac{C}{A} \).

Quartic equation

A quartic equation is a polynomial equation of the fourth degree. Using Ferrari’s Method, it is written in the general form \( x^4 + ax^3 + bx^2 + cx + d = 0 \).

The four roots of this equation (\( x_1, x_2, x_3, x_4 \)) have specific relationships with the coefficients:

• The sum of all four roots (\( x_1 + x_2 + x_3 + x_4 \)) is equal to \( -a \).
• The product of all four roots (\( x_1 x_2 x_3 x_4 \)) is equal to \( d \).
• The sum of the successive products of two roots (\( x_1x_2 + x_2x_3 + x_3x_4 + x_4x_1 + x_1x_3 + x_2x_4 \)) is equal to \( b \).
• The sum of the successive products of three roots (\( x_1x_2x_3 + x_2x_3x_4 + x_1x_2x_4 + x_1x_3x_4 \)) is equal to \( -c \).

Binomial theorem

A binomial is an algebraic expression containing exactly two terms. When raising a binomial \( (x+y) \) to a positive integer power \( n \), the resulting expansion follows strict predictable properties:

• The total number of terms in the resulting expansion is \( n + 1 \).
• The power of the first term \( x \) decreases by 1 in each successive term, while the power of the second term \( y \) increases by 1.
• The sum of the exponents of \( x \) and \( y \) in any single term is always exactly \( n \).
• The expansion begins with \( x^n \) and ends with \( y^n \), both having a numerical coefficient of 1.
• The numerical coefficients increase symmetrically to the middle and then decrease.

The coefficients can be found using Pascal’s Triangle, named after the French mathematician Blaise Pascal. Alternatively, you can calculate the coefficient of any following term based on the preceding term (P.T.) using a reliable ratio: multiply the coefficient of the P.T. by the exponent of \( x \) in the P.T., then divide by the exponent of \( y \) in the P.T. plus 1.

Degree of a polynomial or equation refers to the maximum sum of the exponents of the variables in any single term of the polynomial.

Logarithms

A logarithm is an exponent. The expression \( \log_b x \) represents the exact power to which the base \( b \) must be raised to produce the number \( x \). For example, \( \log_2 16 = 4 \) means the same thing as \( 2^4 = 16 \).

• Common logarithm (denoted as \(\log_{10}\) or simply \(\log\)) uses the decimal base 10. It is also known as Briggsian logarithm.
• Natural logarithm (denoted as \(\ln\)) uses the base \( e \). It is also known as Napierian logarithm.
• Binary logarithm (denoted as \(\text{lb}\)) uses a base value of 2.

To convert between natural logarithms and common logarithms, you use a specific constant called the modulus of logarithms. Multiplying the natural logarithm by 0.4343 gives the common logarithm.

Formulas

• Quadratic formula: \( x = \dfrac{-B \pm \sqrt{B^2 - 4AC}}{2A} \)
• Binomial expansion: \( (x+y)^n = x^n + nx^{n-1}y + \dfrac{n(n-1)}{2!}x^{n-2}y^2 + \dots + nxy^{n-1} + y^n \)
• r-th term of a binomial: \( r^{\text{th}} \text{ term} = {}_{n}C_{r-1} x^{n-r+1} y^{r-1} \)
• Sum of binomial coefficients: \( \text{Sum} = (\text{coefficient of } x + \text{coefficient of } y)^n \)
• Sum of binomial exponents: \( \text{Sum} = n(n+1) \)
• Euler’s Number definition: \( e = \lim_{n\to\infty} \left(1 + \dfrac{1}{n}\right)^n \)
• Logarithm of a product: \( \log(xy) = \log x + \log y \)
• Logarithm of a quotient: \( \log\left(\dfrac{x}{y}\right) = \log x - \log y \)
• Logarithm of a power: \( \log(x^n) = n \log x \)
• Base conversion: \( \log_b x = \dfrac{\log x}{\log b} \)
• Natural to common log: \( \log x = 0.4343 \ln x \)
• Common to natural log: \( \ln x = 2.3026 \log x \)

Quick check

If a quadratic equation has a discriminant equal to exactly zero, what does this tell you about the graph of the function?

Show answer

It means the roots are real and identical. Graphically, the vertex of the parabola touches the x-axis at exactly one point without crossing it.

Why is the logarithm of 1 equal to zero regardless of the base you use?

Show answer

Because any non-zero base raised to the power of zero equals 1 (\( b^0 = 1 \)). Therefore, the exponent needed to get 1 is always zero.