Hiroshima Entrance Math key terms

Writing a positive integer as a product of prime numbers only, for example 84 = 2² × 3 × 7. Used to find square roots, GCDs, and LCMs exactly.

The distance of a number from zero on the number line, always non-negative. Written |x|; equals x when x ≥ 0 and −x when x < 0.

A number that, multiplied by itself, gives the original number. Every positive number has a positive and a negative square root; √a means the non-negative one.

Rewriting a fraction so no square root remains in the denominator, by multiplying numerator and denominator by a suitable root or conjugate, e.g. 1/√2 = √2/2.

Removing brackets by multiplying every term inside by the factor outside, then collecting like terms. The reverse operation is factorization.

Rewriting a polynomial as a product of simpler expressions, such as x² + 5x + 6 = (x + 2)(x + 3). Essential for solving quadratic equations.

Standard expansion identities to memorize: (a+b)² = a²+2ab+b², (a−b)² = a²−2ab+b², (a+b)(a−b) = a²−b², and (x+a)(x+b) = x²+(a+b)x+ab.

A relationship where y = ax for a constant a ≠ 0; doubling x doubles y. The graph is a straight line through the origin.

A relationship where y = a/x for a constant a ≠ 0; as x increases y decreases. The graph is a hyperbola with the axes as asymptotes.

A function of the form y = ax + b with a ≠ 0. Its graph is a straight line with slope a and y-intercept b.

The constant a in y = ax + b, equal to the rise divided by the run between any two points on the line. Parallel lines share the same slope.

The value of y where a graph crosses the y-axis, written b in y = ax + b. Found by setting x = 0.

For a function, the change in y divided by the corresponding change in x. Constant for linear functions, but varying for quadratic functions unless an interval is specified.

Two or more equations in the same variables solved simultaneously. The solution is the pair of values satisfying all equations, found by elimination or substitution.

Replacing a variable with an equal expression or value. Used to solve simultaneous equations and to find intersections of graphs.

Solving simultaneous equations by adding or subtracting the two equations, after scaling if needed, so one variable cancels out.

The U-shaped curve that is the graph of y = ax². It is symmetric about the y-axis and opens upward when a > 0, downward when a < 0.

The vertical line about which a parabola is mirror-symmetric. For y = a(x − p)² + q the axis is the line x = p.

The highest or lowest point of a parabola, lying on its axis of symmetry. For y = a(x − p)² + q the vertex is (p, q).

Rewriting ax² + bx + c in the form a(x − p)² + q so the vertex and axis can be read off directly, and so the quadratic formula can be derived.

An equation of the form ax² + bx + c = 0 with a ≠ 0. It has up to two real solutions, found by factoring, completing the square, or the quadratic formula.

The formula x = (−b ± √(b² − 4ac)) / 2a giving the solutions of ax² + bx + c = 0. It works for every quadratic equation.

The quantity D = b² − 4ac for ax² + bx + c = 0. D > 0 gives two distinct real solutions, D = 0 one repeated solution, D < 0 no real solutions.

Two figures have the same shape when corresponding angles are equal and corresponding side lengths are in a fixed ratio, the scale factor.

The ratio of corresponding side lengths of two similar figures. Areas scale as the square of this ratio and volumes as its cube.

Two figures coincide exactly when moved without resizing: all corresponding sides and angles are equal. Triangles are congruent by SSS, SAS, or ASA with corresponding parts.

Sides that occupy the same relative position in two congruent or similar figures, matching equal angles. Their lengths are equal (congruent) or in a fixed ratio (similar).

Angles on opposite sides of a transversal cutting two parallel lines, between the lines. They are equal when the lines are parallel.

Angles in matching positions at the two intersections made by a transversal cutting two lines. They are equal exactly when the two lines are parallel.

When two parallel lines are cut by a transversal, alternate interior angles are equal and corresponding angles are equal; conversely, equal angles prove the lines parallel.

The three interior angles of any triangle add to 180°. An exterior angle equals the sum of the two non-adjacent interior angles.

The interior angles of an n-sided polygon add to 180(n − 2) degrees. For a quadrilateral this is 360°.

The angle formed between one side of a polygon and the extension of an adjacent side. The exterior angles of any convex polygon sum to 360°.

An angle inscribed in a circle is half the central angle standing on the same arc. Inscribed angles on the same arc are equal, and an angle in a semicircle is 90°.

An angle whose vertex is the centre of a circle, measured by the arc it intercepts. It is twice any inscribed angle standing on the same arc.

A line touching a circle at exactly one point. A tangent is perpendicular to the radius drawn to the point of contact.

In a right triangle with legs a and b and hypotenuse c, a² + b² = c². The converse also holds and is used to prove a triangle right-angled.

The sum of all data values divided by the number of values. Sensitive to extreme values, unlike the median.

The middle value when data are arranged in order; with an even number of values, the average of the two middle ones. Resistant to outliers.

The value or class that occurs most often in a data set. A data set may have no mode or several modes.

The likelihood of an event, between 0 and 1. For equally likely outcomes it equals the number of favourable outcomes divided by the total number of outcomes.

The event that a given event does not happen. Its probability is 1 minus the probability of the event, often the easier route to an answer.

A branching diagram listing all possible outcomes of a sequence of events, used to count cases and compute probabilities without missing any combination.

The count of distinct possible outcomes of a situation, found by systematic listing, tree diagrams, or the multiplication principle for successive choices.