Y-KNOT

Algebra Cheatsheet

Linear Equations

Equations of the form \(ax+b=0\) (one variable) or \(ax+by+c=0\) (two variables). The solution to a system of linear equations is their point of intersection.

Conditions for Solutions (Two Variables)

For a system of two linear equations \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\):

  • Unique Solution: If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \). The lines intersect at one point.
  • No Solution: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \). The lines are parallel.
  • Infinite Solutions: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \). The lines are coincident (the same line).

Quadratic Equations

An equation of the form \(ax^2 + bx + c = 0\), where \(a \neq 0\).

Roots of the Equation (Shreedhara Acharya's Formula)

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

Sum and Product of Roots

For roots \(\alpha\) and \(\beta\):

Sum of roots (\(\alpha + \beta\)) = \(-\frac{b}{a}\)
Product of roots (\(\alpha \beta\)) = \(\frac{c}{a}\)

Nature of Roots (Discriminant, D)

The discriminant, \(D = b^2 - 4ac\), determines the nature of the roots:

  • If \(D > 0\) and is a perfect square: Roots are real, rational, and unequal.
  • If \(D > 0\) and is not a perfect square: Roots are real, irrational, and unequal (conjugate surds).
  • If \(D = 0\): Roots are real, rational, and equal.
  • If \(D < 0\): Roots are complex and unequal (complex conjugates).

Formation of a Quadratic Equation

  • From Roots \(\alpha\) and \(\beta\): \( (x-\alpha)(x-\beta) = 0 \), which expands to \(x^2 - (\alpha+\beta)x + \alpha\beta = 0\).
  • From Sum (S) and Product (P): \( x^2 - Sx + P = 0 \).

Maximum/Minimum Value

  • If \(a > 0\), the parabola opens upwards and has a minimum value of \(\frac{4ac-b^2}{4a}\) at \(x = -\frac{b}{2a}\).
  • If \(a < 0\), the parabola opens downwards and has a maximum value of \(\frac{4ac-b^2}{4a}\) at \(x = -\frac{b}{2a}\).

Inequalities and Modulus

Modulus Properties

The modulus of x, \(|x|\), is the non-negative value of x.

  • If \(|x| \le k\), then \(-k \le x \le k\).
  • If \(|x| \ge k\), then \(x \ge k\) or \(x \le -k\).
  • Triangle Inequality: \(|a| + |b| \ge |a+b|\)
  • \(|a| - |b| \le |a-b|\)

Important Inequalities

  • For any positive real number x, \(x + \frac{1}{x} \ge 2\).
  • Arithmetic Mean \(\ge\) Geometric Mean \(\ge\) Harmonic Mean. For two positive numbers a and b:
$$ \frac{a+b}{2} \ge \sqrt{ab} \ge \frac{2ab}{a+b} $$

Progressions & Series

Arithmetic Progression (A.P.)

  • nth Term: \(T_n = a + (n-1)d\)
  • Sum of n terms: \(S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}[a+l]\)

Geometric Progression (G.P.)

  • nth Term: \(T_n = ar^{n-1}\)
  • Sum of n terms: \(S_n = \frac{a(r^n - 1)}{r-1}\) or \(S_n = \frac{a(1 - r^n)}{1-r}\)
  • Sum of infinite terms (where \(|r| < 1\)): \(S_\infty = \frac{a}{1-r}\)

Harmonic Progression (H.P.)

A sequence is in H.P. if the reciprocals of its terms are in A.P.

  • Harmonic Mean (HM) of a and b = \(\frac{2ab}{a+b}\)
  • Relationship of Means: AM \(\ge\) GM \(\ge\) HM
  • Key Property: \(GM^2 = AM \times HM\)

Logarithms, Surds, and Indices

Indices (Exponents)

  • \(x^m \times x^n = x^{m+n}\)
  • \(\frac{x^m}{x^n} = x^{m-n}\)
  • \((x^m)^n = x^{mn}\)
  • \(x^{-m} = \frac{1}{x^m}\)
  • \(x^{a/b} = \sqrt[b]{x^a}\)

Logarithms

If \(N = a^x\), then \(x = \log_a N\). (\(N > 0, a > 0, a \neq 1\))

  • \(\log_a(xy) = \log_a x + \log_a y\)
  • \(\log_a(\frac{x}{y}) = \log_a x - \log_a y\)
  • \(\log_a(x^n) = n \log_a x\)
  • Change of Base: \(\log_a b = \frac{\log_c b}{\log_c a}\)
  • \(\log_a b = \frac{1}{\log_b a}\)
  • \(a^{\log_a x} = x\)