Indices, surds and quadratics flashcards

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Surd
An irrational root such as √2 or ∛5. Leaving it as a root keeps an answer exact.
Simplest surd form
k√n where n has no square factor other than 1, e.g. √72 = 6√2.
Rationalising the denominator
Rewriting a fraction so the denominator has no surd, by multiplying top and bottom by the same expression.
Conjugate of a + √b
a − √b. Their product (a + √b)(a − √b) = a² − b is rational.
Discriminant of ax² + bx + c
b² − 4ac, the expression under the root in the quadratic formula.
Vertex (turning point) of a parabola
The maximum or minimum point; it lies on the line of symmetry.
Disguised quadratic
An equation of the form a(f)² + b(f) + c = 0 where f is something like √x, x^(1/3), x² or 2^x.
Laws of indices (same base)
aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ. (No — learn it)
Zero and negative indices
a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ. (No — learn it)
Fractional index a^(m/n)
a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ). Take the root first. (No — learn it)
Multiplying and dividing roots
√(ab) = √a √b and √(a/b) = √a/√b. But √(a + b) ≠ √a + √b. (No — learn it)
Completed square form of ax² + bx + c
a(x + b/2a)² + c − b²/4a: vertex at x = −b/2a. (No — derive it)
Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a. (No — learn it)
Nature of roots from b² − 4ac
> 0 two distinct real roots; = 0 one repeated root; < 0 no real roots. (No — learn it)
Shape of y = ax² + bx + c
∪-shaped (minimum) if a > 0; ∩-shaped (maximum) if a < 0.
Where does y = ax² + bx + c cross the axes?
y-axis at (0, c); x-axis at the real roots of ax² + bx + c = 0.
Graph of a quadratic with zero discriminant
The vertex lies on the x-axis: the curve touches the axis at the repeated root.
Vertex and line of symmetry of y = a(x + p)² + q
Vertex (−p, q); line of symmetry x = −p.
Point common to every graph y = xⁿ (x > 0)
(1, 1), since 1ⁿ = 1 for every power n.
Why is a⁰ = 1?
a³ ÷ a³ = a³⁻³ = a⁰, and any non-zero number divided by itself is 1.
Why does multiplying by the conjugate remove the surd?
Difference of two squares: (a + √b)(a − √b) = a² − (√b)² = a² − b, which is rational.
Why is the least value of (x − 3)² + 5 equal to 5?
A square is never negative, so (x − 3)² ≥ 0; the least value 5 occurs when x = 3.
Why does b² − 4ac decide the number of roots?
The formula has ±√(b² − 4ac): two values if positive, one if zero, none (real) if negative.
In a √x substitution, why reject u = √x < 0?
√x means the positive root, so it can never be negative.
Evaluate 8^(−2/3).
8^(1/3) = 2, so 8^(2/3) = 4 and 8^(−2/3) = 1/4.
Simplify √50 + √18.
5√2 + 3√2 = 8√2.
Rationalise 4/(3 + √5).
Multiply by (3 − √5)/(3 − √5): 4(3 − √5)/(9 − 5) = 3 − √5.
Complete the square: x² + 6x + 2.
(x + 3)² − 9 + 2 = (x + 3)² − 7; minimum point (−3, −7).
Write 3x² − 12x + 5 in the form a(x + p)² + q.
3(x² − 4x) + 5 = 3[(x − 2)² − 4] + 5 = 3(x − 2)² − 7.
x² + 6x + k = 0 has equal roots. Find k.
36 − 4k = 0, so k = 9 (and the root is x = −3).
Solve x² − 4x − 1 = 0 exactly.
(x − 2)² = 5, so x = 2 ± √5.
Solve x⁴ − 13x² + 36 = 0.
u = x²: (u − 4)(u − 9) = 0, so x² = 4 or 9: x = ±2, ±3.
“Give your answer in exact form”
Leave surds and fractions; a rounded decimal does not count.
Writing 3/√x before differentiating
3x^(−1/2): the coefficient stays outside; only x takes the power.
Discriminant with an unknown in a or c
Bracket it: b² − 4(k − 1)(k + 2), then expand carefully.
Solving x² = 5x
Never divide by x: x(x − 5) = 0 gives x = 0 or x = 5.
Vertex of (x + 4)² − 7
(−4, −7), not (4, −7): the bracket is zero when x = −4.
Solving 2^(2x+1) − 5 × 2^x + 2 = 0
Split the index first: 2^(2x+1) = 2(2^x)², giving 2u² − 5u + 2 = 0, so u = ½ or 2 and x = −1 or 1.

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