Ukraine 8 клас Algebra
Chapters: 5
1. Algebraic expressions
Rational expressions · Basic property of a rational fraction · Operations on rational fractions · Powers with integer exponents · Square roots · Quadratic trinomial
- Laws of Exponents – An exponent tells how many times a base is multiplied by itself. Same base: multiply → add exponents, divide → subtract, power of a power → multiply. a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) is the nth root.
- Square Roots: Build a Square, Find Its Side – The square root of a number n is the number that, multiplied by itself, gives n. √25 = 5 because 5 × 5 = 25. Picture n tiles arranged in a square: the root is the side. Perfect squares (1, 4, 9, 16, …) have whole-number roots. Other roots are irrational and lie between two whole numbers; we estimate them or simplify them, like √50 = 5√2.
- 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.
2. Numbers
Sets and subsets · Rational, irrational and real numbers
- 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.
3. Equations
Equivalent and rational equations · Quadratic equations · Vieta's theorem · Equations reducible to quadratic
- Rational Equations – A rational (fractional) equation has the unknown in the denominator of a fraction. First write the banned values that make any denominator zero. Then multiply every term by the lowest common denominator (LCD) to clear the fractions, and solve the equation that is left (often linear or quadratic). Finally check each root: a root that is a banned value is extraneous and is thrown out. The same method solves rate problems such as boats on rivers and people working together.
- 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.
4. Functions
Functions y=k/x, y=x² and y=√x · Graphical solving of equations and systems
- Inverse Variation (Inverse Proportion) – Two quantities vary inversely when their product stays the same: x × y = k, so y = k/x. If x is doubled, y is halved. The graph of y = k/x (k > 0) is a curve called a hyperbola that comes close to both axes but never touches them.
- 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.
5. Mathematical tasks and real-world processes
Standard form of a number · Models with rational equations and functions · Story and competence problems
- Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.
- Percentages – Per cent means "out of 100", so x% = x/100. To find x% of an amount, multiply by x/100. Percentage change = (change ÷ original) × 100. To increase by r%, multiply by (1 + r/100); to decrease by r%, multiply by (1 − r/100). Successive changes multiply their multipliers. For reverse percentage, divide the new value by the multiplier.