What's the difference between 'roots', 'solutions', and 'x-intercepts'?

They are closely related concepts. 'Roots' or 'solutions' refer to the algebraic values of $x$ that solve the equation $ax^2+bx+c=0$. 'x-intercepts' are the geometric points where the graph of the function $y=ax^2+bx+c$ crosses the x-axis. The x-coordinates of these intercepts are the real roots of the equation.

When should I use the quadratic formula instead of factorising?

Always try to factorise first, as it's faster. If you can't spot the factors within about 20-30 seconds, or if the question asks for an answer to a specific number of decimal places or significant figures, it's a strong hint that you should use the quadratic formula.

How do I complete the square if the coefficient of x² isn't 1?

If you have $ax^2+bx+c$, you must first factor out the coefficient 'a' from the first two terms, giving $a(x^2 + (b/a)x) + c$. Then, you complete the square inside the bracket as usual and multiply out at the end. See the second worked example for a step-by-step guide.

What does 'no real roots' mean graphically?

Graphically, 'no real roots' means the parabola does not touch or cross the x-axis at all. If the coefficient 'a' is positive, the entire curve will be above the x-axis. If 'a' is negative, the entire curve will be below the x-axis.

Is the vertex always the lowest point of the parabola?

Only when $a > 0$. If $a < 0$ the parabola is n-shaped and the vertex is the highest point (a maximum). The sign of $a$ decides minimum versus maximum every time — try dragging the vertex above the x-axis with a negative $a$ in the explorer to see it flip.

Does a larger value of 'a' make the parabola wider or narrower?

Narrower. Increasing $|a|$ stretches the curve vertically so it rises more steeply and looks narrower; values of $|a|$ between 0 and 1 make it wider and flatter, and a negative $a$ flips it upside-down.

Quadratics 9709 — Free A-Level Maths

Quadratics

Welcome to the world of quadratics! These second-degree polynomials are more than just abstract equations; they model real-world phenomena like the trajectory of a projectile, the shape of a satellite dish, and profit curves in economics. Mastering quadratics is a fundamental skill for A-Level Mathematics, forming the bedrock for many more advanced topics.

BY THE END, YOU CAN…

1Solve quadratic equations by factorisation, completing the square, and using the quadratic formula.
2Use the discriminant, b² - 4ac, to determine the number and nature of the roots of a quadratic equation.
3Express a quadratic function y = ax² + bx + c in the completed square form y = a(x - h)² + k.
4Use the completed square form to find the coordinates of the vertex and sketch the graph of the quadratic function.

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Parabola's Secrets

A quadratic function creates a U-shaped curve called a parabola. By understanding its equation, we can unlock all its key features, like where it crosses the axes and its highest or lowest point.

THINK OF IT LIKE…

Think of throwing a ball. Its path through the air is a perfect parabola. The quadratic equation describes this path, telling us the maximum height the ball reaches (the vertex) and where it will land (the roots).

1
A quadratic equation y = ax² + bx + c has a parabolic graph (a ≠ 0).
2
Completing the square reveals the vertex form y = a(x − h)² + k.
3
The discriminant b² − 4ac determines the nature of roots.
4
Factorise or use the formula to solve ax² + bx + c = 0 on Paper 1.

What this topic covers

The official Cambridge syllabus points this lesson works through.

1.1.1
Complete the square for $ax^2 + bx + c$
Carry out the process of completing the square and use the completed-square form to find the vertex and the least/greatest value.
1.1.2
Find and use the discriminant
Use $b^2 - 4ac$ to determine the number and nature of the roots of $ax^2 + bx + c = 0$ .
1.1.3
Solve quadratic equations and quadratic inequalities
Solve quadratics in one unknown by factorising, completing the square or the formula, and solve quadratic inequalities.
1.1.4
Solve a linear–quadratic simultaneous system
Solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic.
1.1.5
Recognise equations that are quadratic in a function of $x$
Recognise and solve equations such as $x^4 - 5x^2 + 4 = 0$ that are quadratic in some function of $x$ .

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

INTERACTIVESTEP 1 / 4 · Shape

Change a — the parabola stretches, and flips when a is negative.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$ax^2 + bx + c = 0$

Tap a symbol — great for exam definitions

$y = a(x-h)^2 + k$

Tap a symbol — great for exam definitions

At a glance — side by side

Compare key properties side by side — ideal for exam contrasts.

Three ways to solve $ax^2 + bx + c = 0$ — pick the right tool

Method	Reach for it when…	Always works?	Bonus it gives you
Factorisation	The factors jump out within ~20 seconds	No — only if it factorises neatly	The fastest marks in the exam
Completing the square	You also need the vertex, or it is a 'show that' proof	Yes	The vertex $(h, k)$ directly
Quadratic formula	Factors will not come, or an answer to 3 s.f. is asked	Yes	The discriminant $b^2 - 4ac$

Factorisation

Reach for it when…

The factors jump out within ~20 seconds

Always works?

No — only if it factorises neatly

Bonus it gives you

The fastest marks in the exam

Completing the square

Reach for it when…

You also need the vertex, or it is a 'show that' proof

Always works?

Yes

Bonus it gives you

The vertex

(h, k)

directly

Quadratic formula

Reach for it when…

Factors will not come, or an answer to 3 s.f. is asked

Always works?

Yes

Bonus it gives you

The discriminant

b^2 - 4ac

Full topic notes

Formal explanation with the rigour you need for the exam.

Solving Quadratic Equations

A key task is to find the 'roots' or 'solutions' of a quadratic equation, which are the values of $x$ that satisfy $ax^2 + bx + c = 0$ . These roots correspond to the x-intercepts of the graph of $y = ax^2 + bx + c$ . There are three primary methods for solving these equations.

The Quadratic Formula: For $ax^2 + bx + c = 0$ , the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Factorisation: If the quadratic expression can be written as a product of two linear factors, $(px+q)(rx+s)=0$ , then the solutions are $x = -q/p$ and $x = -s/r$ . This is the quickest method when it's possible.
Completing the Square: This method rewrites the equation to isolate $x$ . It's a reliable method that also helps in finding the vertex of the parabola.
The Quadratic Formula: This formula works for any quadratic equation and is essential when factorisation is not straightforward. It is derived from the method of completing the square.

The Discriminant: Understanding the Roots

The expression inside the square root of the quadratic formula, $b^2 - 4ac$ , is called the discriminant. Its value tells us about the number and type of roots the equation has, without needing to solve the equation fully. This is a powerful tool for analysing quadratic functions.

If $\boldsymbol{b^2 - 4ac > 0}$ , there are two distinct real roots. The parabola crosses the x-axis twice.
If $\boldsymbol{b^2 - 4ac = 0}$ , there is one repeated real root. The parabola touches the x-axis at its vertex.
If $\boldsymbol{b^2 - 4ac < 0}$ , there are no real roots. The parabola is entirely above or entirely below the x-axis.

EXAM TIP

Exam questions frequently ask for the set of values of a constant, say $k$ , for which a quadratic equation has 'real roots' or 'two distinct roots'. For 'real roots', use the condition $b^2 - 4ac \ge 0$ . For 'two distinct roots', use the strict inequality $b^2 - 4ac > 0$ . Be careful with the wording!

Completing the Square and Finding the Vertex

Completing the square is a technique for rewriting a quadratic of the form $ax^2 + bx + c$ into the vertex form $a(x-h)^2 + k$ . This form is incredibly useful as it tells us the coordinates of the turning point (vertex) of the parabola, which is $(h, k)$ , and whether it's a maximum or a minimum.

Vertex Form: $y = a(x-h)^2 + k$ \ Vertex at $(h, k)$

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

Solve the equation $3x^2 - 7x - 5 = 0$ , giving your answers correct to 3 significant figures.

1
This quadratic does not factorise easily, so we use the quadratic formula. Here, $a=3$ , $b=-7$ , and $c=-5$ .

EXAMPLE 2

The equation $x^2 + (k+3)x + (k+6) = 0$ has two distinct real roots. Find the set of values of $k$ .

1
'Two distinct real roots' means we need the discriminant to be strictly positive: $b^2 - 4ac > 0$ .

EXAMPLE 3

Express $f(x) = 2x^2 + 12x + 5$ in the form $a(x+p)^2 + q$ . Hence, state the coordinates of the vertex and whether it is a maximum or minimum point.

1
First, factor out the coefficient of $x^2$ from the first two terms: $f(x) = 2(x^2 + 6x) + 5$

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAQuadratics

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 12

What is the general form of a quadratic equation?

TAP OR SPACE TO FLIP

ANSWER

$ax^2 + bx + c = 0$ , where $a, b, c$ are constants and $a \neq 0$ .

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Factorisation: If the quadratic expression can be written as a product of two linear factors, $(px+q)(rx+s)=0$ , then the solutions are $x = -q/p$ and $x = -s/r$ . This is the quickest method when it's possible.
✓
Completing the Square: This method rewrites the equation to isolate $x$ . It's a reliable method that also helps in finding the vertex of the parabola.
✓
The Quadratic Formula: This formula works for any quadratic equation and is essential when factorisation is not straightforward. It is derived from the method of completing the square.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Test Your Knowledge on Quadratics

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Khan Academy: Quadratics↗

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Test Your Knowledge on Quadratics on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

NEXT →1.2 Functions

Quadratics

BY THE END, YOU CAN…

1Solve quadratic equations by factorisation, completing the square, and using the quadratic formula.
2Use the discriminant, b² - 4ac, to determine the number and nature of the roots of a quadratic equation.
3Express a quadratic function y = ax² + bx + c in the completed square form y = a(x - h)² + k.
4Use the completed square form to find the coordinates of the vertex and sketch the graph of the quadratic function.

Method

Reach for it when…

Always works?

Bonus it gives you

Factorisation

The factors jump out within ~20 seconds

No — only if it factorises neatly

The fastest marks in the exam

Completing the square

You also need the vertex, or it is a 'show that' proof

Yes

The vertex

(h, k)

directly

Quadratic formula

Factors will not come, or an answer to 3 s.f. is asked

Yes

The discriminant

b^2 - 4ac

Revise Quadratics for Cambridge 9709 (1.1)

In simple terms

What this topic covers

Explore the concept

Key formulas

At a glance — side by side

Factorisation

Completing the square

Quadratic formula

Full topic notes

Solving Quadratic Equations

The Discriminant: Understanding the Roots

Completing the Square and Finding the Vertex

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Test Your Knowledge on Quadratics

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

What this topic covers

Explore the concept

Key formulas

At a glance — side by side

Factorisation

Completing the square

Quadratic formula

Full topic notes

Solving Quadratic Equations

The Discriminant: Understanding the Roots

Completing the Square and Finding the Vertex

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Test Your Knowledge on Quadratics

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Quadratics