Algebra & Quadratics

The index laws, fractional & negative powers, simplifying surds and rationalising denominators. ✎

① The Index Laws

aᵐ × aⁿ = a m+n

aᵐ ÷ aⁿ = a m−n

(aᵐ)ⁿ = a mn

(ab)ⁿ = aⁿbⁿ

Zero: a⁰ = 1 (a ≠ 0)

Fractional:

n √ a

② Reading a Fractional Power

= (∛8)² = 2² =

Take the root first — the numbers stay small. (∛8)² = 4 is far easier than ∛64.

③ Simplifying Surds

√ ab = √ a × √ b

√ a/b = √ a ÷ √ b

Pull out the largest square factor:

√ 50 = √ 25 × 2 = 5√ 2 √ 72 = √ 36 × 2 = 6√ 2

Like surds then add like terms: 5√ 2 + 6√ 2 = 11√ 2.

④ Rationalising the Denominator

Clear the surd from the bottom by multiplying top & bottom by the same thing:

SINGLE SURD — multiply by √a

1 / √ a = √ a / a

TWO TERMS — multiply by the CONJUGATE

1 / a + √ b → × a − √ b / a − √ b

The bottom becomes a² − b — no surd left.

Flip the sign only — the conjugate of 3 + √2 is 3 − √2 (not −3 − √2).

★ Worked example ✎

Write 6 / √ 3 − 1 in the form a + b√3.

The denominator has two terms, so multiply top and bottom by its conjugate √3 + 1. This is multiplying by 1, so the value is unchanged.

6 / √ 3 − 1 × √ 3 + 1 / √ 3 + 1

Combine into a single fraction.

= 6(√ 3 + 1) / (√ 3 − 1)(√ 3 + 1)

Expand the denominator — it is a difference of two squares, so the surd terms cancel.

(√3 − 1)(√3 + 1) = 3 + √3 − √3 − 1 = 2

Expand the numerator.

6(√3 + 1) = 6√3 + 6

Divide every term on the top by 2.

= 6√3 + 6 / 2 = 3√3 + 3

Write in the requested order a + b√3 and state the values.

3 + 3√3 so a = 3 and b = 3

Exam trap

Expand the bottom fully before cancelling. Students often write the answer over √3 − 1 still — no marks for a surd left downstairs.

A-LEVEL MATHEMATICS · EDEXCEL 9MA0 · PURE YEAR 1 — CH 2

Quadratics

Solving, completing the square, the discriminant & sketching — plus hidden quadratics. ✎

① Three Ways to Solve

All start from ax² + bx + c = 0, a ≠ 0:

1 · FACTORISE

x² − 5x + 6 = (x − 2)(x − 3) = 0 → x = 2 or 3

2 · QUADRATIC FORMULA formula booklet

x = −b ± √ b² − 4ac / 2a

3 · COMPLETE THE SQUARE

Best when the question asks for the turning point or an exact answer.

② Completing the Square

x² + bx + c = (x + b / 2 )² − ( b / 2 )² + c

Written as a(x + p)² + q, you can read off:

Turning point (−p, q)

Line of symmetry x = −p

(x + 3)² − 8

→ minimum at (−3, −8)

③ The Discriminant

Δ = b² − 4ac

Its sign tells you how many times the curve meets the x-axis:

two distinct real roots

Δ = 0

one repeated root (tangent)

no real roots

"Show it has no real roots " → prove Δ < 0. " Two distinct roots" → set up Δ > 0 and solve the resulting inequality in k.

④ Hidden Quadratics

Substitute to reveal a quadratic, solve, then convert back:

x⁴ − 5x² + 4 = 0 → let y = x² → y² − 5y + 4 = 0

x − 5√x + 6 = 0 → let y = √x → y² − 5y + 6 = 0

2x

y = 2ˣ

The equation x² + (k + 3)x + 4k = 0 has two equal roots. Find the possible values of k.

"Two equal roots" means the discriminant is zero, so the condition to use is:

b² − 4ac = 0

Read off the coefficients from the equation, keeping the bracket intact.

a = 1, b = (k + 3), c = 4k

Substitute into the condition.

(k + 3)² − 4(1)(4k) = 0

Expand the bracket and simplify to a quadratic in k.

k² + 6k + 9 − 16k = 0 k² − 10k + 9 = 0

Factorise and solve (two numbers multiplying to 9 and adding to −10: −1 and −9).

(k − 1)(k − 9) = 0

State both solutions.

k = 1 or k = 9

Bracket b before squaring. Writing k + 3² instead of (k + 3)² loses every mark that follows. And note b² − 4ac is in the formula booklet, but you must identify a, b, c yourself.