Year 11 in England

10th grade algebra

19 10th grade algebra skills, each with a worked example, free practice that checks every answer, and a printable worksheet. Start with a mixed review, or jump to the skill you need.

Mixed review

Ten questions drawn from every skill below.

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The quadratic formula

CCSS HSA.REI.B.4.BEngland: GCSE Higher Algebra

Example

Solve 3x² − 6x − 3 = 0 using the quadratic formula. Give the larger solution to 2 decimal places.

b² − 4ac = 36 − (−36) = 72
x = (6 ± √72) / 6
x ≈ 2.41 or x ≈ −0.41

Answer: 2.41

Builds on: Solve quadratics by factorising, Simplify surds (radicals)

Leads to: Complex numbers, Use the discriminant to find unknown coefficients, Give exact solutions with the quadratic formula

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Completing the square

CCSS HSA.SSE.B.3.BCCSS HSA.REI.B.4.AEngland: GCSE Higher Algebra

Example

Write x² − 4x − 11 in the form (x + p)² + q. What is q?

p = −4 ÷ 2 = −2
(x − 2)² = x² − 4x + 4
q = −11 − 4 = −15

Answer: -15

Builds on: Multiply binomials (expand double brackets), Graphs of quadratics: roots, vertex, intercepts

Leads to: Equation of a circle, Quadratic functions in vertex form, Find the centre and radius from x² + y² + Dx + Ey + F = 0, Solve quadratics by completing the square (exact answers)

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Factorise ax² + bx + c

CCSS HSA.SSE.A.2England: GCSE Higher Algebra

Example

Factorise: 5x² − 8x − 21

(5x + 7)(x − 3)

Answer: (5x + 7)(x − 3)

Builds on: Factorise quadratics x² + bx + c

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Quadratic inequalities

CCSS HSA.REI.B.4England: GCSE Higher Algebra

Example

Solve: x² − 8x + 12 > 0

(x − 2)(x − 6) > 0
x < 2 or x > 6

Answer: x < 2 or x > 6

Builds on: Solve quadratics by factorising, Solve linear inequalities

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Linear and quadratic systems

CCSS HSA.REI.C.7England: GCSE Higher Algebra

Builds on: Systems of linear equations (simultaneous equations), Solve quadratics by factorising

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Algebraic fractions

CCSS HSA.APR.D.6CCSS HSA.APR.D.7England: GCSE Higher Algebra

Example

Simplify: (x² − 12x + 35) / (x − 7)

(x − 7)(x − 5) / (x − 7)
x − 5

Answer: x − 5

Builds on: Factorise quadratics x² + bx + c, Add and subtract fractions with unlike denominators

Leads to: Rational equations, Add and subtract algebraic fractions

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Composite and inverse functions

CCSS HSF.BF.A.1.CCCSS HSF.BF.B.4England: GCSE Higher Algebra

Example

f(x) = 5x − 4 and g(x) = x² + 1. Find f(g(3)).

g(3) = 3² + 1 = 10
f(10) = 5 × 10 − 4 = 46

Answer: 46

Builds on: Function notation, Rearrange formulas (change the subject)

Leads to: Inverse functions (non-linear)

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Fractional and negative indices

CCSS HSN.RN.A.1CCSS HSN.RN.A.2England: GCSE Higher Number

Example

Evaluate 81^(1/4).

4√81 = 3
3¹ = 3

Answer: 3

Builds on: Laws of exponents (indices), Square roots and cube roots

Leads to: Logarithms, Evaluate fractional and negative fractional powers, Rewrite roots as fractional powers and back, Solve exponential equations by matching bases

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Solve quadratics by completing the square (exact answers)

CCSS HSA.REI.B.4.ACCSS HSA.REI.B.4.BEngland: GCSE Higher Algebra

Example

Rewrite x² + 8x − 7 = 0 as (x + 4)² = k. Find k.

(x + 4)² − 16 − 7 = 0
k = 23

Answer: 23

Builds on: Completing the square, Solve quadratics by taking square roots

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Use the discriminant to find unknown coefficients

CCSS HSA.REI.B.4.BEngland: GCSE Higher AlgebraEngland: A-level Pure Mathematics

Example

Work out the discriminant of 2x² + 8x + 6 = 0.

8² − 4 × 2 × 6
= 16

Answer: 16

Builds on: The quadratic formula

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Give exact solutions with the quadratic formula

CCSS HSA.REI.B.4.BEngland: GCSE Higher Algebra

Example

Solve 2x² − 3 = 0. Give exact answers in the form (p ± √q)/r.

b² − 4ac = 24
x = (0 ± √24)/4

Answer: x = (0 ± √24)/4

Builds on: The quadratic formula, Simplify surds (radicals)

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Evaluate fractional and negative fractional powers

CCSS HSN.RN.A.1CCSS HSN.RN.A.2England: GCSE Higher Number

Example

Evaluate 16^(1/4).

4√16 = 2

Answer: 2

Builds on: Fractional and negative indices, Evaluate powers with zero and negative exponents

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Rewrite roots as fractional powers and back

CCSS HSN.RN.A.2England: GCSE Higher Algebra

Example

Write ∛(x²) as a power of x.

x^(2/3)

Answer: x^(2/3)

Builds on: Fractional and negative indices

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Evaluate composite functions f(g(a))

CCSS HSF.BF.A.1.CEngland: GCSE Higher Algebra

Example

f(x) = −3x + 4 and g(x) = x² − 2. Find g(f(−4)).

f(−4) = 16
g(16) = 254

Answer: 254

Builds on: Function notation

Leads to: Write the rule for a composite function

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Write the rule for a composite function

CCSS HSF.BF.A.1.CEngland: GCSE Higher Algebra

Example

f(x) = −x + 6 and g(x) = 3x + 5. Find f(g(x)) and simplify.

−(3x + 5) + 6
−3x + 1

Answer: -3x+1

Builds on: Evaluate composite functions f(g(a))

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Find the inverse of a linear function

CCSS HSF.BF.B.4.AEngland: GCSE Higher Algebra

Example

f(x) = −6x + 6. Find f⁻¹(x).

x = −6y + 6
f⁻¹(x) = (x − 6)/(−6)

Answer: (x-6)/(-6)

Builds on: Rearrange formulas (change the subject), Function notation

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Find domains of functions with roots and fractions

CCSS HSF.IF.B.5CCSS HSF.IF.A.1England: A-level Pure Mathematics

Example

What is the domain of f(x) = √(2 − x) (real numbers only)?

x ≤ 2

Answer: x ≤ 2

Builds on: Function notation, Solve linear inequalities

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Simplify algebraic fractions by factorising

CCSS HSA.APR.D.6CCSS HSA.SSE.A.2England: GCSE Higher Algebra

Example

Simplify: (x² + 7x + 12)/(x + 3)

(x + 3)(x + 4)/(x + 3)
(x + 4)

Answer: (x + 4)

Builds on: Factorise quadratics x² + bx + c, Difference of two squares

Leads to: Multiply and divide algebraic fractions, Evaluate limits by substitution or by factorising

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Multiply and divide algebraic fractions

CCSS HSA.APR.D.7England: GCSE Higher Algebra

Example

Simplify: (x² − 64)/(x − 7) × (x − 7)/(x − 8)

(x + 8)(x − 8)(x − 7)/((x − 7)(x − 8))
(x + 8)

Answer: (x + 8)

Builds on: Simplify algebraic fractions by factorising

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