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Algebra and Functions

Indices, Surds and Quadratics

Pearson Edexcel A-level Mathematics

Laws of indices

LawForm
Multiplyaxay ≡ ax + y
Divideax ÷ ay ≡ ax − y
Power of a power(ax)y ≡ axy
Fractional indexam/n = n√am = (n√a)m
Negative indexa−n = 1an

Surds

RuleForm
Product√xy = √x√y
Square(√x)2 = x
Conjugate pair(√x + √y)(√x − √y) = x − y
Method: simplifying a surd
  1. Write the number under the root as the largest square factor times what remains.
  2. Split the root with the product rule.
  3. Take the root of the square factor outside.
  4. Write the answer as k√m, with no square factor left in m.
  • When a "simplified surd" is asked for, the unsimplified root is never the final answer.
  • To rationalise a denominator, multiply numerator and denominator by the conjugate of the denominator: √a + √b by √a − √b, and √a by √a.
Worked example: a length as a simplified surd

The points P and Q have coordinates (1, −2) and (7, 10). Find the exact length of PQ, writing your answer as a fully simplified surd.

PQ = √(7 − 1)2 + (10 + 2)2 = √36 + 144 = √180

√180 = √36 × √5

PQ = 6√5

Completing the square

Method: completing the square
  1. Take the coefficient of x2 out of the x2 and x terms only.
  2. Inside the bracket, halve the coefficient of x to get b, and write (x + b)2 − b2.
  3. Multiply back out by the coefficient and collect the constants.
  4. Write the answer as the full identity, f(x) = a(x + b)2 + c.
  • The turning point of y = a(x + b)2 + c is (−b, c).
  • It is a minimum when a > 0 and a maximum when a < 0.
Worked example: completing the square with a negative coefficient

f(x) = 5 + 18x − 3x2

(a) Write f(x) in the form a(x + b)2 + c, where a, b and c are integers to be found.

(b) Hence state the coordinates of the maximum point of the curve with equation y = f(x).

(a) f(x) = −3(x2 − 6x) + 5

= −3[(x − 3)2 − 9] + 5 = −3(x − 3)2 + 27 + 5

f(x) = −3(x − 3)2 + 32

(b) (3, 32)

The discriminant

ax2 + bx + c = 0 has roots x = −b ± √b2 − 4ac2a

b2 − 4acRootsLine and curve
> 0two distinct real rootsmeet at two distinct points
= 0one repeated rootthe line is a tangent
< 0no real rootsdo not meet
Method: a line and a curve, range of k
  1. Substitute the line into the curve.
  2. Collect to ax2 + bx + c = 0, with a, b and c in terms of k.
  3. Apply the condition on b2 − 4ac.
  4. Solve the resulting quadratic in k for the critical values.
  5. Choose the region for k as for any quadratic inequality.
Worked example: range of k for a line that does not meet a curve

The line with equation y = kx − 3, where k is a constant, does not meet the curve with equation y = x2 + 2x + 1. Find the set of possible values of k, writing your answer in set notation.

x2 + 2x + 1 = kx − 3

x2 + (2 − k)x + 4 = 0

b2 − 4ac < 0 ⇒ (2 − k)2 − 4 × 1 × 4 < 0

k2 − 4k − 12 < 0 ⇒ (k + 2)(k − 6) < 0

Critical values k = −2, k = 6

{k : −2 < k < 6}

Trap: the sign of the x term

Collecting to x2 + (k − 2)x + 4 = 0 still gives the right critical values, because b is squared, but the equation is wrong. Write x2 + (2 − k)x + 4 = 0.

To show a quadratic factor has no real roots, calculate its discriminant and state b2 − 4ac < 0, so the quadratic has no real roots.

Quadratics in a function of x

  • A quadratic in sin x, ex, ln x or a power of x: substitute a single letter, solve the quadratic, then substitute back.
  • Reject any root the function cannot take, and say why: ex > 0, −1 ≤ sin x ≤ 1.

These notes are part of an Excel with Osman subscription and are not available to print.

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