Flashcards · J560 · Topic 6

Algebra flashcards

39 cards on Topic 6, subtopics 6.01 to 6.06. Pick a subtopic or study them all.

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Like terms: Terms with the same letters to the same powers. 4ab and −ba are like terms; 3x and 3x² are not. Keep the sign with its term: 6x + 2y − x + 4y = 5x + 6y.
Expand and simplify (x + 5)(x − 3): Four products: x² − 3x + 5x − 15 = x² + 2x − 15.
(a + b)² and (a + b)(a − b): (a + b)² = a² + 2ab + b² (not a² + b²). (a + b)(a − b) = a² − b².
How do you factorise x² + bx + c?: Find two numbers with product c and sum b, then write (x + m)(x + n). If c is negative the numbers have different signs. Expand to check.
Factorise fully 3x² − 27: Common factor first: 3(x² − 9). Then the difference of two squares: 3(x − 3)(x + 3).
Prove that the sum of two consecutive odd numbers is a multiple of 4Higher: (2n + 1) + (2n + 3) = 4n + 4 = 4(n + 1), which is 4 times an integer. Examples are never a proof.
Factorise 8x² + 10x − 3Higher: ac = −24; two numbers with product −24 and sum 10: 12 and −2. 8x² + 12x − 2x − 3 = 4x(2x + 3) − 1(2x + 3) = (4x − 1)(2x + 3).
Write x² − 8x + 3 in the form (x + p)² + qHigher: Halve −8 to get −4: (x − 4)² − 16 + 3 = (x − 4)² − 13.
Simplify (x² + x − 12)/(x² − 16)Higher: Factorise: (x + 4)(x − 3) / ((x − 4)(x + 4)). Cancel the factor (x + 4): (x − 3)/(x − 4). Cancel whole factors, never single terms.

Expression, formula and equation: Expression: no equals sign. Formula: links real quantities, such as v = u + at. Equation: true only for certain values; you solve it.
Substitute a = −2, b = −5 into 3a² − b: Use brackets: 3(−2)² − (−5) = 3(4) + 5 = 17. Note (−2)² = 4 but −2² = −4.
How do you change the subject when it appears twice?: Clear fractions, expand, collect every term with the new subject on one side, factorise it out, then divide.
Kinematics formulae for constant acceleration: v = u + at s = ut + ½at² v² = u² + 2as u initial velocity, v final velocity, a acceleration, s displacement, t time.
Kinematics: what do "starts from rest" and "stops" tell you?: Starts from rest: u = 0. Stops: v = 0. Choose the formula with exactly one unknown.

Solve 5x + 2 = 3x + 14: Subtract 3x: 2x + 2 = 14. Subtract 2: 2x = 12. x = 6. Check in the original equation.
Solve (x − 2)(x + 5) = 0: Each bracket equals 0: x = 2 or x = −5. The roots have the opposite signs to the numbers in the brackets.
Why is "x² = 9x, so x = 9" wrong?: Dividing by x loses a root. Rearrange and factorise: x² − 9x = 0, x(x − 9) = 0, so x = 0 or x = 9.
How do you solve linear simultaneous equations by elimination?: 1. Multiply so one unknown has equal coefficients. 2. Same signs: subtract; opposite signs: add. 3. Solve, substitute into an original equation. 4. Check in the other equation.
Quadratic formula and the discriminantHigher: x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0. b² − 4ac > 0: two real solutions; = 0: one repeated; < 0: no real solutions.
Solve x² + 2x − 7 = 0 by completing the squareHigher: (x + 1)² − 1 − 7 = 0, so (x + 1)² = 8. x = −1 ± √8 = −1 ± 2√2
Solve y = 9 − x and xy = 20 simultaneouslyHigher: Substitute: x(9 − x) = 20, so x² − 9x + 20 = 0, (x − 4)(x − 5) = 0. x = 4, y = 5 or x = 5, y = 4. Find each partner from the linear equation.
Sign-change method for f(x) = 0Higher: If f(a) and f(b) have opposite signs, a root lies between a and b, provided f is continuous on that interval. Narrow it by decimal search or interval bisection.

Solve 3(x − 2) ≥ 5x + 4: 3x − 6 ≥ 5x + 4 −2x ≥ 10 x ≤ −5 (dividing by −2 reverses the sign)
Solve −5 < 2x − 1 ≤ 9: Do the same to all three parts. Add 1: −4 < 2x ≤ 10. Divide by 2: −2 < x ≤ 5.
Number line: open or closed circle?: Open circle for < or > (boundary not included). Closed circle for ≤ or ≥ (boundary included).
Solve x² − x − 6 > 0Higher: (x − 3)(x + 2) = 0 gives critical values −2 and 3. A U-shaped curve is above the axis outside the roots: x < −2 or x > 3 (two separate inequalities).
Set notation: what does {x | x ≥ 3} mean?Higher: The set of all x such that x ≥ 3. The bar means "such that".
How do you show an inequality as a region on a graph?Higher: Draw the boundary line: solid for ≤ or ≥, dashed for < or >. Test a point not on the line, such as (0, 0), to find which side satisfies the inequality.

The function machine y = 2x + 3: input −1, and the input for output 11: Input −1: 2(−1) + 3 = 1. Output 11: 2x + 3 = 11, so x = 4.
How do you find the inverse of a function machine?Higher: Reverse the order of the steps and replace each with its opposite. y = (x − 4)/5 (− 4, then ÷ 5) has inverse × 5, then + 4: x = 5y + 4.
Composite functions: machine C multiplies by 4, machine D subtracts 3Higher: C then D: 4x − 3 (input 6 gives 21). D then C: 4(x − 3) (input 6 gives 12). The order changes the result.
What happens when you reverse a squaring step?Higher: You get both a positive and a negative square root.

Term-to-term rule compared with position-to-term rule: Term-to-term: builds each term from the one before (needs a starting term). Position-to-term: a formula for the nth term.
nth term of 11, 18, 25, 32, …: d = 7, and c = first term − d = 11 − 7 = 4. nth term = 7n + 4.
nth term of 30, 26, 22, 18, …: d = −4, c = 30 − (−4) = 34. nth term = 34 − 4n. Its first negative term is the 9th: 34 − 36 = −2.
Fibonacci-type sequence with 3rd term 11 and 5th term 31. Find it.: Terms a, b, a + b, a + 2b, 2a + 3b. So a + b = 11 and 2a + 3b = 31, giving b = 9, a = 2. Sequence: 2, 9, 11, 20, 31.
Arithmetic compared with geometric sequences: Arithmetic: add the same amount d each time (3, 6, 9). Geometric: multiply by the same ratio r each time (3, 6, 12). Find r by dividing a term by the one before.
Find the nth term of 7, 12, 19, 28, 39, …Higher: Second difference 2, so a = 1. Subtract n²: 6, 8, 10, 12, 14 = 2n + 4. nth term = n² + 2n + 4.
xₙ₊₁ compared with xₙ + 1Higher: xₙ₊₁ is the next term after xₙ. xₙ + 1 is the current term plus one. Example: if xₙ₊₁ = 2xₙ − 3 and x₃ = 21, then x₂ = (21 + 3)/2 = 12.