Quadratic Equations: formula sheet

Key formulas

Standard form

ax² + bx + c = 0, a ≠ 0

a is the coefficient of x², b of x, c is the constant. Always keep signs with the coefficients.

Degree test

Highest power of x = 2 after simplification

Expand brackets and clear fractions before checking the degree.

Number of roots

A quadratic equation has at most 2 roots

The roots may be two different values, two equal values, or not real.

Root check

x = k is a root if a·k² + b·k + c = 0

Substitute the value into the equation to test it.

Consecutive integers

x, x + 1 for integers; x, x + 2 for consecutive even or odd integers

Useful for framing word problems on numbers.

Standard form

ax² + bx + c = 0, a ≠ 0

Bring all terms to the left side so the right side is 0 before you solve.

Zero product rule

If (x − p)(x − q) = 0, then x = p or x = q

Works only when the product equals zero.

Factorisation by splitting the middle term

Find two numbers m and n with m + n = b and m × n = a × c

Split bx into mx + nx, then group terms and take out common factors.

Quadratic formula

x = [−b ± √(b² − 4ac)] ÷ 2a

Works for every quadratic. Put brackets round negative values of b.

Completing the square

x² + bx = (x + b/2)² − (b/2)²

Divide the equation by a first so the coefficient of x² is 1.

Check of roots

Sum of roots = −b ÷ a; product of roots = c ÷ a

Use this to verify your answers quickly.

Discriminant

D = b² − 4ac

For ax² + bx + c = 0 with a ≠ 0. Always write the equation in standard form first.

Real and distinct roots

D > 0

Two different real roots. If D is also a perfect square (rational coefficients), the roots are rational.

Real and equal roots

D = 0

Each root is −b ÷ 2a. The condition is b² = 4ac.

Imaginary roots

D < 0

No real roots. Roots are complex conjugates.

Real roots (either type)

D ≥ 0

Use when a question says only 'roots are real'.

Roots by formula

x = [−b ± √D] ÷ 2a

The discriminant sits under the square root.

Sum of roots

α + β = −b ÷ a

For ax² + bx + c = 0. Mind the minus sign.

Product of roots

αβ = c ÷ a

No minus sign. Always divide by a.

Form equation from roots

x² − (α + β)x + αβ = 0

Use when the equation is to be monic. Multiply by any number for other forms.

Sum of squares

α² + β² = (α + β)² − 2αβ

Most tested identity.

Difference of roots

(α − β)² = (α + β)² − 4αβ

Take the square root at the end; α − β = ±√(...).

Sum of cubes

α³ + β³ = (α + β)³ − 3αβ(α + β)

Also α³ + β³ = (α + β)(α² − αβ + β²).

Sum of reciprocals

1/α + 1/β = (α + β) ÷ αβ

Valid when αβ ≠ 0.

Quick revision

• Standard form: ax² + bx + c = 0, with a ≠ 0.
• Roots are the values of x that satisfy the equation.
• Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a.
• Discriminant D = b² − 4ac.
• D > 0: two distinct real roots. D = 0: equal real roots. D < 0: no real roots.
• For rational coefficients, if D is a perfect square, the roots are rational.
• Sum of roots = −b/a.
• Product of roots = c/a.
• Equation from roots: x² − (sum)x + (product) = 0.
• α² + β² = (α + β)² − 2αβ.
• Roots are reciprocals of each other when c = a.
• Roots are equal in magnitude and opposite in sign when b = 0.

Common mistakes

• Calling x(x + 3) = x² + 5 a quadratic equation. Fix: Expand and move all terms to one side. Here x² + 3x = x² + 5 gives 3x - 5 = 0, which is linear.
• Losing the sign of b or c. Fix: First write the equation as ax² + bx + c = 0. For x² = 5x - 6, it becomes x² - 5x + 6 = 0, so b = -5 and c = 6.
• Solving before moving all terms to one side, for example writing x(x − 5) = 6 as x = 6 or x − 5 = 6. Fix: Expand first: x² − 5x − 6 = 0. Then factorise to (x − 6)(x + 1) = 0, so x = 6 or −1.
• Giving the wrong signs to the roots, for example writing x = 2 from the bracket (x + 2). Fix: Always write x + 2 = 0 and then x = −2.
• Taking the wrong signs of a, b or c Fix: Write a = , b = , c = with signs before using the formula. If x is missing, b = 0.
• Writing D = b² − 4ac as b² − 4 + ac or (b − 4ac)² Fix: Remember it as 'b squared minus four a c'. Compute b² and 4ac separately.
• Writing the sum of roots as b/a instead of −b/a. Fix: Always write S = −b/a first. For x² − 5x + 6 = 0, b = −5, so S = 5.
• Forgetting to divide by a when a ≠ 1. Fix: For 2x² − 8x + 6 = 0, S = 8/2 = 4 and P = 6/2 = 3.

Exam tips

• For 'is it quadratic' questions, always expand first. The x² terms often cancel in the trap options.
• In framing questions, check the options for the standard form and match signs of b and c carefully.
• If options give possible roots, substitute them into the equation. This is often faster than solving.
• Write consecutive numbers correctly: x, x + 1 for integers and x, x + 2 for even or odd integers.
• Do not leave any question blank. There is no negative marking, so make an informed guess after elimination.
• If the equation factorises, do it by factorisation. It is faster than the formula and has fewer sign errors.
• Before solving, check whether the equation is given in disguise, such as a fraction equation or a bracket product. Clear it to standard form first.
• In MCQs, use sum and product of roots to eliminate options before doing full working.