GCSE Maths
Algebra and Equations Flashcards
37 GCSE algebra terms examiners expect you to define, not just use
37 cards~9 min
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All 37 GCSE Maths Algebra and Equations flashcards
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- Algebraic Expression
- A collection of letters and numbers joined by operations, with no equals sign anywhere in it. Because nothing is being equated, an expression can be simplified or evaluated but never solved.For example: 4a + 7b − 2 is an expression; 4a + 7b − 2 = 9 is not.
- Algebraic Fraction
- A fraction whose top or bottom contains letters. It is simplified by factorising both parts first and cancelling any bracket appearing in each, exactly as numerical fractions cancel shared factors.For example: Both parts of (x² − 9) ÷ (x + 3) factorise, leaving x − 3.
- Algebraic Proof
- An argument using letters rather than examples to show a statement holds in every case. Marks come from choosing general expressions and finishing with a written conclusion, not from testing numbers.For example: Writing consecutive integers as n and n + 1 covers every possible pair at once.
- Arithmetic Sequence
- A sequence in which the same amount is added or subtracted each time, so the gap between neighbouring entries never changes. Its rule always takes the form of that common gap multiplied by n, plus a number.For example: 3, 8, 13, 18 climbs by 5 each time, so its rule is 5n − 2.
- Balance Method
- The rule that whatever operation you apply to one side of an equation you must apply identically to the other, keeping the two sides equal while the unknown is gradually isolated.For example: From 5y − 2 = 18, add 2 to both sides, then divide both sides by 5.
- Common Factor
- A number or letter that divides exactly into every term of an expression. Taking the highest one outside a bracket is the first move examiners look for before any other method.For example: 6ab + 9a share 3a, giving 3a(2b + 3).
- Completed Square Form
- A quadratic rewritten as a squared bracket plus a number, in the shape (x + p)² + q. The rewrite exposes the graph's turning point and gives an exact route to the solutions.For example: x² + 6x + 1 becomes (x + 3)² − 8, since (x + 3)² already supplies 9.
- Counterexample
- One specific case that makes a claim false, which is all that is needed to disprove it. Examples can never prove a claim true, but a single one can settle that it is untrue.For example: Against the claim that every prime is odd, quoting 2 finishes the argument.
- Difference of Two Squares
- The pattern in which one square is subtracted from another, which always splits into two brackets holding the same pair of terms with opposite signs.For example: p² − 64 becomes (p + 8)(p − 8), because 64 is 8 squared.
- Discriminant
- The part under the square root sign, b² − 4ac, whose sign tells you how many real solutions a quadratic has: two when positive, one repeated when zero, and none when negative.For example: For x² + 2x + 5 it equals 4 − 20 = −16, so the curve never crosses the x-axis.
- Elimination Method
- A way of handling two equations by matching the numbers in front of one unknown, then adding or subtracting the equations so that unknown disappears and a single-letter equation is left.For example: Doubling one equation to give 4b in both lets you subtract and remove b entirely.
- Fibonacci-Type Sequence
- A sequence in which each entry is found by adding the two entries immediately before it, so two starting values are needed before it can be continued.For example: Starting 1, 3 gives 1, 3, 4, 7, 11, 18.
- Formula
- A rule written in algebra that connects two or more quantities, letting you calculate one of them once the others are known. It is stated as an equality and reused with different numbers.For example: The area of a trapezium, A = ½(a + b)h, links four quantities.
- Fractional Index
- A power written as a fraction, where the bottom number gives the root to take and the top number gives the power to raise the result to.For example: 27^(2/3) means the cube root of 27 squared, which is 3² = 9.
- Function Notation
- The shorthand in which a rule is named and the input written in brackets after it, so that writing a number inside the brackets means substituting that number into the rule.For example: If f(x) = 3x + 1 then f(4) = 13, because 4 replaces x.
- Geometric Sequence
- A sequence built by multiplying by the same number each time, called the common ratio. Dividing any entry by the one before it returns that ratio.For example: 2, 6, 18, 54 has a common ratio of 3.
- Identity
- A statement that stays true whatever numbers the letters stand for, so it has infinitely many solutions rather than a few. Examiners write it with the three-bar symbol instead of an equals sign.For example: 2(n + 5) ≡ 2n + 10 holds for every value of n.
- Index
- The small raised number recording how many times a base is multiplied by itself. Its plural is indices, and the laws governing them let powers be combined without writing the multiplication out.For example: In m⁴ the base is m and the index is 4, meaning m × m × m × m.
- Inverse Function
- The rule that reverses another rule, taking an output back to the input it came from. It is found by writing the rule as an equation and making the input the subject.For example: If f(x) = 3x + 1 then the inverse is (x − 1) ÷ 3.
- Inverse Operation
- The operation that undoes another one, used to isolate a letter during rearranging. Addition and subtraction undo each other, as do multiplication and division, and squaring and taking the square root.For example: To undo +9 you subtract 9; to undo ×4 you divide by 4.
- Iteration
- A method of approximating a solution by feeding a starting estimate into a rearranged rule, then feeding each answer back in repeatedly until the digits settle to the required accuracy.For example: Higher-tier papers give the rearranged rule and a starting value, then ask for three cycles.
- Like Terms
- Terms carrying exactly the same combination of letters raised to exactly the same powers. Only these may be added or subtracted together, because the letter part stays untouched while the numbers in front combine.For example: 7pq and −2pq are like terms; 7pq and 7p²q are not.
- Negative Index
- A power written with a minus sign, which instructs you to take the reciprocal of the positive power rather than to produce a negative answer.For example: 4⁻² means 1 ÷ 4², which is one sixteenth, not −16.
- Nth Term
- A rule in terms of the position number n that generates any entry of a sequence directly, without needing the entries before it. Substituting the position returns that entry.For example: With the rule 5n − 2, position 10 gives 48 immediately.
- Parabola
- The smooth U-shaped curve every quadratic graph makes. It opens upwards when the number in front of the squared term is positive and downwards when that number is negative.For example: y = −x² + 4 is a parabola opening downwards with its peak at (0, 4).
- Position-to-Term Rule
- Another name for a rule that converts a position number straight into the entry sitting at that position, in contrast to a rule that works from the previous entry onwards.For example: Position 1 goes in, the first entry comes out; position 50 goes in, the fiftieth comes out.
- Quadratic Formula
- The general solution x = (−b ± √(b² − 4ac)) ÷ 2a, which works on any quadratic written as ax² + bx + c = 0. It is the standard method when no bracket pair can be found.For example: For 2x² + 3x − 4 = 0 read off a = 2, b = 3 and c = −4 before substituting.
- Quadratic Sequence
- A sequence whose gaps between entries are themselves changing, but whose gaps between those gaps stay the same. Its rule contains an n² term, and half the constant second difference gives the number in front of n².For example: 2, 8, 18, 32 has second differences of 4, so the rule starts with 2n².
- Root
- A value of the unknown that makes an equation true, and on a graph the place where the curve meets the horizontal axis. Roots are also called solutions or zeros.For example: The roots of (x − 1)(x + 6) = 0 are 1 and −6, where each bracket becomes zero.
- Simultaneous Equations
- Two or more equations in two or more unknowns that must all hold at once, so the answer is the pair of values satisfying every equation rather than just one of them.For example: Graphically, the answer is the point where the two lines cross.
- Solution Set
- The complete collection of values that satisfy a statement. For an inequality it is usually an infinite stretch of the number line, described by a range rather than listed one by one.For example: For −2 ≤ n < 5 with n an integer, the set is −2, −1, 0, 1, 2, 3 and 4.
- Strict Inequality
- A comparison using the symbols for less than or greater than only, which excludes the boundary value itself. It is drawn on a number line with an open, unshaded circle.For example: n > 3 excludes 3, so the smallest whole number it allows is 4.
- Subject of a Formula
- The single letter standing alone on one side of a formula, with everything else on the other side. Changing the subject means rearranging so a different letter is the one left by itself.For example: In v = u + at the subject is v; rearranged for a it becomes a = (v − u)/t.
- Substitution Method
- A way of handling two equations by rearranging one to make a letter the subject, then replacing that letter in the other equation so only a single unknown remains.For example: It is the usual choice when one equation is already in the form y = something.
- Term
- One separate part of an expression, made of numbers and letters multiplied together and separated from its neighbours by a plus or minus sign.For example: 5x² − 3xy + 8 has three terms: 5x², −3xy and 8.
- Term-to-Term Rule
- A rule describing how to get from one entry of a sequence to the next one. It needs a starting value to be useful and becomes slow for entries far along the sequence.For example: Start at 4 and add 6 each time gives 4, 10, 16, 22 and so on.
- Turning Point
- The single place on a quadratic graph where it stops falling and starts rising, or the reverse. It is the lowest point of a positive curve and the highest point of a negative one.For example: y = (x − 4)² + 3 turns at the coordinates (4, 3).