Algebra Guide & Practice

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1. Fundamentals & Definitions

• Linear Equation: An algebraic equation where the maximum power (degree) of any variable is unity (1). The general form of a linear equation in two variables is $Ax + By + C = 0$, where $A \neq 0$ and $B \neq 0$.
• System of Linear Equations (Simultaneous Equations): A collection of two or more linear equations sharing common variables.
• Consistent System: A system of simultaneous linear equations that has at least one set of values for the variables that satisfies all equations (i.e., it has a unique solution or infinitely many solutions).
• Inconsistent System: A system of simultaneous linear equations that has no solution.
• Homogeneous System: A system of equations where the constant terms are all zero, represented in two variables as: $$a_1x + b_1y = 0$$ $$a_2x + b_2y = 0$$
• Quadratic Equation: An algebraic equation of the second degree. The general form is $ax^2 + bx + c = 0$, where $a, b, c$ are coefficients (real or complex numbers) and $a \neq 0$.
• Roots (or Solutions) of a Quadratic Equation: The values of the variable $x$ that satisfy the equation. By the Fundamental Theorem of Algebra, a quadratic equation has exactly two roots (which may be real and distinct, real and equal, or complex conjugates).
• Discriminant ($D$): For a quadratic equation $ax^2 + bx + c = 0$, the expression $D = b^2 - 4ac$ is the discriminant. It determines the nature of the roots.
• Linear Inequality (Inequation): An algebraic statement involving variables and inequality symbols ($<$, $>$, $\le$, or $\ge$) instead of an equality sign ($=$).
• Open Half-Plane: The region of the Cartesian plane on one side of a boundary line $ax + by + c = 0$, excluding the line itself. It is represented by $ax + by + c > 0$ or $ax + by + c < 0$.
• Closed Half-Plane: The region of the Cartesian plane on one side of a boundary line $ax + by + c = 0$, including the line itself. It is represented by $ax + by + c \ge 0$ or $ax + by + c \le 0$.
• Complex Conjugates: Two complex numbers of the form $u + iv$ and $u - iv$ (where $i = \sqrt{-1}$). In quadratic equations with real coefficients, complex roots always occur in conjugate pairs.
• Irrational Conjugates: Two irrational numbers of the form $p + \sqrt{q}$ and $p - \sqrt{q}$ (where $q$ is not a perfect square). In quadratic equations with rational coefficients, irrational roots always occur in conjugate pairs.
• Arithmetic Mean (AM) & Geometric Mean (GM): For $n$ positive numbers, the Arithmetic Mean is $\frac{a_1 + a_2 + \dots + a_n}{n}$ and the Geometric Mean is $(a_1 a_2 \dots a_n)^{1/n}$.

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2. Core Concepts & Formulas

Algebraic Identities

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Systems of Linear Equations (Two Variables)

For the simultaneous system: $$a_1x + b_1y = c_1$$ $$a_2x + b_2y = c_2$$

The consistency and solutions are determined by comparing coefficient ratios:

Homogeneous Systems

For a homogeneous system ($a_1x + b_1y = 0$ and $a_2x + b_2y = 0$):

• If $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the system has only the trivial solution: $x = 0, y = 0$.
• If $\frac{a_1}{a_2} = \frac{b_1}{b_2}$, the system has a non-zero solution, resulting in infinitely many solutions.

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Quadratic Equations

For a quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$:

1. Quadratic Formula & Discriminant

The roots are calculated using: $$x = \frac{-b \pm \sqrt{D}}{2a}$$ where $D = b^2 - 4ac$ is the discriminant.

2. Nature of Roots

3. Relation between Roots ($\alpha, \beta$) and Coefficients

• Sum of Roots ($S$): $\alpha + \beta = -\frac{b}{a}$
• Product of Roots ($P$): $\alpha\beta = \frac{c}{a}$
• Difference of Roots: $|\alpha - \beta| = \frac{\sqrt{D}}{|a|}$
• Equation Construction: $x^2 - Sx + P = 0 \implies x^2 - (\alpha + \beta)x + \alpha\beta = 0$

4. Commonly Used Symmetric Relations of Roots

• $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = S^2 - 2P$
• $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) = S^3 - 3SP$
• $\alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2\alpha^2\beta^2 = (S^2 - 2P)^2 - 2P^2$
• $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{S}{P}$
• $\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{\alpha^2\beta^2} = \frac{S^2 - 2P}{P^2}$

5. Special Parameter Conditions

• Reciprocal Roots: If roots are reciprocals of each other ($\alpha = \frac{1}{\beta}$), then $c = a$.
• Equal Magnitude, Opposite Sign: If $\alpha = -\beta$, then $b = 0$.
• One Root is Zero: If $c = 0$, one root is $0$.
• One Root is Square of the Other: If $\alpha = \beta^2$, then $b^3 + ac^2 + a^2c = 3abc$.
• Roots in Ratio $m : n$: If $\alpha : \beta = m : n$, then $ac(m+n)^2 = mn b^2$.
• One Root is $k$ More Than the Other: If $\alpha - \beta = k$, then $b^2 - 4ac = k^2 a^2$.

6. Common Roots Conditions

Let the two quadratic equations be $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$.

• Condition for One Common Root ($\alpha$): $$(c_1a_2 - c_2a_1)^2 = (a_1b_2 - a_2b_1)(b_1c_2 - b_2c_1)$$ The common root is given by: $$\alpha = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1} = \frac{b_1c_2 - b_2c_1}{c_1a_2 - c_2a_1}$$
• Condition for Both Roots Common: $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$

7. Maximum/Minimum Value of $f(x) = ax^2 + bx + c$

• If $a > 0$, the expression has a minimum value of $\frac{4ac - b^2}{4a} = -\frac{D}{4a}$ at $x = -\frac{b}{2a}$.
• If $a < 0$, the expression has a maximum value of $\frac{4ac - b^2}{4a} = -\frac{D}{4a}$ at $x = -\frac{b}{2a}$.

8. Sign of a Quadratic Expression

• $ax^2 + bx + c > 0$ for all real $x$ if $a > 0$ and $D < 0$.
• $ax^2 + bx + c < 0$ for all real $x$ if $a < 0$ and $D < 0$.

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Inequalities

• Basic Properties:
  • If $a > b \implies a + c > b + c$ and $a - c > b - c$.
  • If $a > b$ and $c > 0 \implies ac > bc$ and $\frac{a}{c} > \frac{b}{c}$.
  • If $a > b$ and $c < 0 \implies ac < bc$ and $\frac{a}{c} < \frac{b}{c}$ (the sign of inequality reverses).
  • If $a > b > 0 \implies \frac{1}{a} < \frac{1}{b}$.
  • If $x > 0$ and $a > b > 0 \implies a^x > b^x$.
  • If $a > 1$ and $x > y > 0 \implies a^x > a^y$.
  • If $0 < a < 1$ and $x > y > 0 \implies a^x < a^y$.
• AM-GM Inequality: For any positive real numbers $a_1, a_2, \dots, a_n$: $$\frac{a_1 + a_2 + \dots + a_n}{n} \ge (a_1 a_2 \dots a_n)^{1/n}$$ Equality holds if and only if $a_1 = a_2 = \dots = a_n$.
• Quadratic Inequalities: Let $\alpha$ and $\beta$ be the real roots of $ax^2 + bx + c = 0$ with $\alpha < \beta$ and $a > 0$:
  • $(x - \alpha)(x - \beta) < 0 \implies \alpha < x < \beta$ (solution lies between the roots).
  • $(x - \alpha)(x - \beta) > 0 \implies x < \alpha$ or $x > \beta$ (solution lies outside the roots).
• Absolute Value Inequalities:
  • $|x| \le a \implies -a \le x \le a$
  • $|x| \ge a \implies x \le -a$ or $x \ge a$

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Typical Exam Weightage

A high-weightage SSC topic — identity-based shortcuts (without full expansion) are essential for speed here.

Figures are typical ranges based on recent-year patterns, not a guarantee for any specific upcoming paper — always cross-check against the latest official syllabus and previous-year papers for Algebra.