Quadratic Equation

1. Polynomial

A polynomial is an algebraic expression in which the powers of the variable(s) are non-negative integers.

The general form of a polynomial of degree $n$ in variable $x$ is:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \dots + a_2 x^2 + a_1 x + a_0$

where $a_n, a_{n-1}, \dots, a_0$ are real numbers (coefficients) and $a_n \neq 0$.

Classification of Polynomials by Degree

1.1 Polynomial Equation

A polynomial equation is formed when a polynomial is set equal to zero:

$a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0, \quad a_n \neq 0$

The values of $x$ that satisfy this equation are called the roots (or zeros) of the polynomial.

Fundamental Theorem of Algebra: A polynomial equation of degree $n$ has exactly $n$ roots (counting multiplicities) in the complex number system.

1.2 Quadratic Equation

A quadratic equation is a polynomial equation of degree 2:

$ax^2 + bx + c = 0, \quad a \neq 0$

where $a, b, c$ are real constants.

Methods to Solve a Quadratic Equation

Factorization Method: Express as $(x - p)(x - q) = 0$, then $x = p$ or $x = q$

Completing the Square: Rewrite as $(x - h)^2 = k$, then solve for $x$

Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula works for all quadratic equations.

1.3 Nature of Roots (Discriminant)

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is:

$D = b^2 - 4ac$

The discriminant determines the nature of the roots:

Case 1: $D > 0$ (Positive)

• Two distinct real roots
• Roots are: $x = \frac{-b \pm \sqrt{D}}{2a}$
• Graph intersects the x-axis at two distinct points

Case 2: $D = 0$ (Zero)

• Two equal real roots (repeated root)
• Root is: $x = \frac{-b}{2a}$
• Graph touches the x-axis at exactly one point

Case 3: $D < 0$ (Negative)

• Two complex conjugate roots
• No real roots exist
• Graph does not intersect the x-axis

Summary Table

2.1 Relation Between Roots and Coefficients

Let $\alpha$ and $\beta$ be the roots of the quadratic equation:

$ax^2 + bx + c = 0, \quad a \neq 0$

Then the quadratic equation can be written as:

$a(x - \alpha)(x - \beta) = 0$

Expanding:

$a[x^2 - (\alpha + \beta)x + \alpha\beta] = 0$

$ax^2 - a(\alpha + \beta)x + a\alpha\beta = 0$

Comparing with $ax^2 + bx + c = 0$, we get the relations:

a. Sum of Roots

$\alpha + \beta = -\frac{b}{a}$

b. Product of Roots

$\alpha\beta = \frac{c}{a}$

2.2 Formation of a Quadratic Equation

If $\alpha$ and $\beta$ are the roots of a quadratic equation, then the quadratic equation is:

$x^2 - (\alpha + \beta)x + \alpha\beta = 0$

Or equivalently:

$x^2 - Sx + P = 0$

where $S = \alpha + \beta$ (sum of roots) and $P = \alpha\beta$ (product of roots).

Steps to Form a Quadratic Equation:

1. Find the sum of roots: $S = \alpha + \beta$
2. Find the product of roots: $P = \alpha\beta$
3. Write the equation: $x^2 - Sx + P = 0$

2.3 Symmetric Functions of Roots

A symmetric function of roots $\alpha$ and $\beta$ is a function that remains unchanged when $\alpha$ and $\beta$ are interchanged.

Important Symmetric Expressions:

(1) Sum of Squares

$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$

(2) Sum of Cubes

$\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$

(3) Sum of Fourth Powers

$\alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha\beta)^2$

(4) Difference of Roots

$(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$

$\alpha - \beta = \pm\sqrt{(\alpha + \beta)^2 - 4\alpha\beta}$

(5) Sum of Reciprocals

$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta}$

(6) Sum of Squares of Reciprocals

$\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{(\alpha\beta)^2}$

(7) $\alpha^2\beta + \alpha\beta^2$

$\alpha^2\beta + \alpha\beta^2 = \alpha\beta(\alpha + \beta)$

(8) $\alpha^3 + \beta^3$ (Alternative Form)

$\alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2)$

3.1 Common Root of Two Quadratic Equations

Consider two quadratic equations:

$a_1x^2 + b_1x + c_1 = 0, \quad a_1 \neq 0$ $a_2x^2 + b_2x + c_2 = 0, \quad a_2 \neq 0$

Let $\alpha$ be a common root of both equations.

Then $\alpha$ satisfies both equations:

$a_1\alpha^2 + b_1\alpha + c_1 = 0$ $a_2\alpha^2 + b_2\alpha + c_2 = 0$

3.2 Condition for One Common Root

If the two quadratic equations have exactly one common root $\alpha$, then:

$\frac{\alpha^2}{b_1c_2 - b_2c_1} = \frac{\alpha}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}$

From this, the common root is:

$\alpha = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1} = \frac{c_1a_2 - c_2a_1}{b_1c_2 - b_2c_1}$

Condition for One Common Root

Eliminating $\alpha$ from the above relations:

$(b_1c_2 - b_2c_1)(a_1b_2 - a_2b_1) = (c_1a_2 - c_2a_1)^2$

3.3 Condition for Both Roots Common

If the two quadratic equations have both roots common, then they are identical (or proportional):

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

That is:

$a_1b_2 - a_2b_1 = 0, \quad b_1c_2 - b_2c_1 = 0, \quad c_1a_2 - c_2a_1 = 0$

3.4 Condition for Both Roots Common (Alternate Form)

Two quadratic equations have both roots common if and only if:

$a_1b_2 - a_2b_1 = 0 \quad \text{and} \quad b_1c_2 - b_2c_1 = 0$

3.5 Common Root Formula Summary

Key Formula Summary

1. Quadratic Equation

$ax^2 + bx + c = 0, \quad a \neq 0$

2. Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

3. Discriminant

$D = b^2 - 4ac$

4. Sum and Product of Roots

$\alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}$

5. Formation of Quadratic Equation

$x^2 - (\alpha + \beta)x + \alpha\beta = 0$

6. Symmetric Functions

$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$

$\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$

$(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$

7. Common Roots

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \quad \text{(both roots common)}$

$(b_1c_2 - b_2c_1)(a_1b_2 - a_2b_1) = (c_1a_2 - c_2a_1)^2 \quad \text{(one root common)}$