Russia 9 класс Algebra (basic)

Chapters: 4

1. Numbers and calculations

Real numbers · Approximations

Irrational Numbers: √2, √3, the Square Root Spiral and Real Numbers – An irrational number cannot be written as p/q. Its decimal never ends and never repeats, like √2 = 1.41421356… or π. We can still mark √2 exactly on the number line: it is the diagonal of a square of side 1. We prove √2 and √3 are irrational by assuming they are fractions in lowest terms and reaching a contradiction. The square root spiral builds √2, √3, √4, √5 … one right triangle at a time. Rational and irrational numbers together form the real numbers; every repeating decimal can be turned back into p/q.

Equations in one variable · Systems of equations · Inequalities

Quadratic Equations – A quadratic equation has x² as its highest power: ax² + bx + c = 0 with a ≠ 0. It has at most two roots. Find them by factorising (split the middle term, then set each bracket to zero) or by the formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac tells the nature of roots: D > 0 two different real roots, D = 0 two equal roots, D < 0 no real roots.
Pair of Linear Equations in Two Variables – Two equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ each make a straight line. The answer that fits both is the point where the lines meet. Lines that cross give one answer, parallel lines give none, and lines that lie on top of each other give endless answers. We can find the answer by drawing (graph), by substitution or by elimination.
Quadratic Inequalities – A quadratic inequality asks where ax² + bx + c is above zero (> 0) or below zero (< 0). Find the roots, picture the parabola, and read the answer from the graph. When a > 0 the curve is below zero between the roots and above zero outside them. If there are no real roots (D < 0), the curve is always on one side of the x-axis.

Quadratic function · Graphs of basic functions

Quadratic Functions and Their Graphs – A quadratic function is y = ax² + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola. If a > 0 it opens up and has a lowest point; if a < 0 it opens down and has a highest point. That turning point is the vertex, at x = −b/2a. In vertex form y = a(x − h)² + k the vertex is (h, k).
Graphs of Basic Functions and Solving Equations with Graphs – A graph shows every point (x, y) with y = f(x). Five shapes come up again and again: the parabola y = x², the cubic y = x³, the half-parabola y = √x, the V-shape y = |x| and the hyperbola y = k/x. Learn each one's domain, range, symmetry, where it rises or falls, and where it crosses the axes. Then use graphs to solve equations (where graphs cross) and inequalities (where one graph is above the other).

Sequences · Arithmetic and geometric progressions

Sequences and Progressions – A sequence is a list of numbers in a fixed order. We can describe it with a recursive rule (how to get the next term from the last one) or an explicit rule (a formula for term n). In an arithmetic progression (AP) we add the same number every time. In a geometric progression (GP) we multiply by the same number every time. Fractals and the Tower of Hanoi are fun patterns that hide these rules.
Arithmetic Progressions – An arithmetic progression (AP) is a list of numbers where each term is made by adding the same fixed number d (the common difference) to the term before. With first term a: nth term aₙ = a + (n − 1)d; sum of the first n terms Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.