Edexcel IAL P1 Algebra: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

P1 algebra is the toolkit for the rest of the course: indices, surds, quadratics and the discriminant, simultaneous equations and inequalities. Most questions want an exact answer, so keep surds and fractions until the last line. The discriminant $b^{2} - 4 a c$ tells you how many real roots a quadratic has without solving it.

P1 · Pure Mathematics 1
Calculator allowed
3 worked examples
119 practice questions
Tested in 4 of the last 4 papers

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Notes
Formulas
Worked examples
Mistakes
FAQ

P1 Algebra notes

Indices

Multiply powers of the same base by adding the indices, divide by subtracting them, and raise a power to a power by multiplying: $a^{m} \times a^{n} = a^{m + n}$ , $a^{m} \div a^{n} = a^{m - n}$ , $(a^{m})^{n} = a^{m n}$ . A negative index is a reciprocal and a fractional index is a root: $8^{\frac{2}{3}} = 4$ and $16^{- \frac{1}{2}} = \frac{1}{4}$ .

To solve an equation with the unknown in the power, write both sides as powers of the same base and equate the indices: $4^{x} = 32$ becomes $2^{2 x} = 2^{5}$ , so $x = \frac{5}{2}$ .

Surds

Simplify a surd by taking out the largest square factor: $\sqrt{50} = 5 \sqrt{2}$ . To rationalise a denominator of the form $a - \sqrt{b}$ , multiply top and bottom by $a + \sqrt{b}$ , because $(a - \sqrt{b}) (a + \sqrt{b}) = a^{2} - b$ has no surd. For example $\frac{4}{3 - \sqrt{5}} = 3 + \sqrt{5}$ .

Quadratics and completing the square

Solve a quadratic by factorising, by the quadratic formula or by completing the square. Completing the square also gives the turning point: $x^{2} - 6 x + 5 = (x - 3)^{2} - 4$ , so the graph of $y = x^{2} - 6 x + 5$ has its minimum point at $(3, - 4)$ . When the $x^{2}$ coefficient is not 1, take it out of the $x$ terms first.

The discriminant

For $a x^{2} + b x + c = 0$ : $b^{2} - 4 a c > 0$ gives two distinct real roots, $b^{2} - 4 a c = 0$ one repeated root, and $b^{2} - 4 a c < 0$ no real roots. Questions usually hide a constant $k$ in the coefficients. If $k x^{2} + 4 x + 1 = 0$ has equal roots, then $16 - 4 k = 0$ , so $k = 4$ .

Simultaneous equations

With one linear and one quadratic equation, make $x$ or $y$ the subject of the linear one and substitute into the quadratic. From $y = x + 1$ and $x^{2} + y^{2} = 13$ : $x^{2} + (x + 1)^{2} = 13$ , so $x^{2} + x - 6 = 0$ , giving $x = 2, y = 3$ or $x = - 3, y = - 2$ . Pair each $x$ with its own $y$ .

Linear and quadratic inequalities

Solve a linear inequality like an equation, but reverse the sign when you multiply or divide by a negative number. For a quadratic inequality, find the critical values and sketch the parabola: $x^{2} - x - 6 = (x - 3) (x + 2)$ , so $x^{2} - x - 6 < 0$ for $- 2 < x < 3$ , and $x^{2} - x - 6 > 0$ for $x < - 2$ or $x > 3$ . Set notation such as ${x : - 2 < x < 3}$ is also accepted.

Key formulas

Quadratic formula Learn it: $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
Discriminant $b^{2} - 4 a c$ : $> 0$ two real roots, $= 0$ equal roots, $< 0$ no real roots Learn it
Completing the square Learn it: $x^{2} + b x + c = {(x + \frac{b}{2})}^{2} + c - \frac{b^{2}}{4}$
Indices; surds Learn it: $a^{- n} = \frac{1}{a^{n}}, a^{\frac{m}{n}} = \sqrt[n]{a^{m}}; \frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P1 formulas on one page (free PDF) →

P1 Algebra worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 7 marks · hard

Let $f (x) = 5 - 4 x - 2 x^{2}$ .

(a) Express $f (x)$ in the form $A - B (x + C)^{2}$ , where $A$ , $B$ , and $C$ are integers.
(b) State the maximum value of $f (x)$ and the value of $x$ at which it occurs.
(c) Solve $f (x) = 0$ , giving your answers in exact form.

Mark scheme

M1 Completes the square inside the bracket: $5 - 2 ((x + 1)^{2} - 1)$ .
A1 Expands and simplifies to obtain $7 - 2 (x + 1)^{2}$ .

B1 States this occurs at $x = - 1$ .

A1 Solves for $x$ by square rooting, yielding $x = - 1 \pm \sqrt{\frac{7}{2}}$ (or equivalent exact surd).

7 steps · 1 to fill in

Worked example 2 · 7 marks · hard

Solve the simultaneous equations $x - y = 2$ and $x^{2} - x y + y^{2} = 7$ .

Mark scheme

M1 Substitutes this expression into the quadratic equation: $x^{2} - x (x - 2) + (x - 2)^{2} = 7$ .
M1 Expands the brackets correctly: $x^{2} - x^{2} + 2 x + (x^{2} - 4 x + 4) = 7$ .
A1 Simplifies to a 3-term quadratic equation: $x^{2} - 2 x - 3 = 0$ .
M1 Factorises the quadratic: $(x - 3) (x + 1) = 0$ .
A1 Finds the $x$ -values: $x = 3$ and $x = - 1$ .
A1 Substitutes back into the linear equation to find the corresponding $y$ -values, giving the final solution pairs $x = 3, y = 1$ and $x = - 1, y = - 3$ .

7 steps · 1 to fill in

Worked example 3 · 5 marks · medium

The quadratic equation $x^{2} + k x + (k + 3) = 0$ , where $k$ is a constant, has equal roots. Find the possible values of $k$ .

Mark scheme

A1 Substitutes the coefficients to form an equation in $k$ : $k^{2} - 4 (1) (k + 3) = 0$ .
M1 Expands to a 3-term quadratic: $k^{2} - 4 k - 12 = 0$ .
M1 Factorises the quadratic equation: $(k - 6) (k + 2) = 0$ .
A1 States the possible values for $k$ are $k = 6 or k = - 2$ .

5 steps

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Common P1 algebra mistakes

Splitting a root over a sum. $\sqrt{9 + 16} = 5$ , not $3 + 4 = 7$ . Only products split: $\sqrt{a b} = \sqrt{a} \sqrt{b}$ .
Not reversing an inequality. Dividing $- 2 x > 6$ by $- 2$ gives $x < - 3$ . Keep the sign the same and the answer is the wrong half of the number line.
One interval where there should be two. $x^{2} > 9$ means $x < - 3$ or $x > 3$ . Writing $x > \pm 3$ does not earn the final mark.
Unpaired solutions. In simultaneous equations each value of x has its own value of y. Give them as pairs (or coordinates), not as two separate lists.
Squaring −(k + 3) without brackets. For $x^{2} - (k + 3) x + 4 = 0$ , $b = - (k + 3)$ and $b^{2} = (k + 3)^{2}$ . Two distinct real roots need $(k + 3)^{2} - 16 > 0$ , so $k + 3 > 4$ or $k + 3 < - 4$ : $k > 1$ or $k < - 7$ . Use $>$ here, not $\geq$ , which would also allow equal roots.

Every P1 mistake from the examiner reports, with the paper it came from →

On your calculator: how to find the roots and vertex of a quadratic. A calculator is allowed in every IAL unit.

P1 Algebra FAQ

Do I need to learn the quadratic formula for IAL P1?

Yes. On our IAL formula sheet the quadratic formula and the discriminant conditions are both marked Learn it, so know them by heart.

What is the discriminant used for in P1?

It tells you how many real roots ax² + bx + c = 0 has without solving it: two if b² − 4ac is positive, one repeated root if it is zero and none if it is negative. A typical question gives a quadratic with a constant k and asks for the values of k that give a certain number of roots, or shows that a line is a tangent to a curve.

Is a calculator allowed in IAL P1?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P1 included. Method marks are for the working you write down, so show it.

How long is the IAL P1 exam?

Each IAL maths unit, P1 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does algebra come up in P1?

In our mapping of the 4 Pearson P1 papers from June 2024 to June 2025, algebra was tested in 4 of them (12 questions in all). Any topic can be examined in any series.

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More P1: Coordinate Geometry · Trigonometry · Differentiation · Integration · Graphs and Transformations

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Other units: IAL P2 · IAL P3 · IAL P4 · IAL S1 · IAL M1

Edexcel IAL P1 Algebra: exam questions and answers, notes and worked examples

Try it first

P1 Algebra notes

Indices

Surds

Quadratics and completing the square

The discriminant

Simultaneous equations

Linear and quadratic inequalities

Key formulas

P1 Algebra worked examples

Worked example 1 · 7 marks · hard

Mark scheme

Worked example 2 · 7 marks · hard

Mark scheme

Worked example 3 · 5 marks · medium

Mark scheme

Common P1 algebra mistakes

P1 Algebra FAQ

Do I need to learn the quadratic formula for IAL P1?

What is the discriminant used for in P1?

Is a calculator allowed in IAL P1?

How long is the IAL P1 exam?

How often does algebra come up in P1?

Next steps

Your P1 revision kit