Equations: formula sheet

Key formulas

Standard form

ax + b = 0, a ≠ 0

Every linear equation in one variable reduces to this form.

Solution

x = −b ÷ a

Gives the single root. It needs a ≠ 0.

Balance rule

Do the same operation on both sides

Never multiply or divide by zero. Dividing by an expression that can be zero may lose or add roots.

Consecutive numbers

x, x + 1, x + 2 (integers); x, x + 2, x + 4 (odd or even)

Use x − 1, x, x + 1 or x − 2, x, x + 2 when the sum or middle value is asked.

Digits of a two-digit number

Number = 10t + u

t is the tens digit and u is the units digit. The reversed number is 10u + t.

Age after or before n years

Age now ± n

Add n for the future, subtract n for the past. Apply it to every person in the problem.

Money value

Total value = number of items × value of each item

Keep the units (₹ or paise) the same on both sides.

Standard form (two unknowns)

a₁x + b₁y = c₁ and a₂x + b₂y = c₂

Bring both equations to this form before using any method. Keep signs with the coefficients.

Cross-multiplication result

x = (c₁b₂ − c₂b₁) ÷ (a₁b₂ − a₂b₁) and y = (a₁c₂ − a₂c₁) ÷ (a₁b₂ − a₂b₁)

Valid only when a₁b₂ − a₂b₁ ≠ 0. The denominator is the same for both x and y.

Unique solution test

a₁ ÷ a₂ ≠ b₁ ÷ b₂

The lines cross at exactly one point. Use cross-multiplication or elimination.

No solution test

a₁ ÷ a₂ = b₁ ÷ b₂ ≠ c₁ ÷ c₂

The lines are parallel. The system is inconsistent.

Infinite solutions test

a₁ ÷ a₂ = b₁ ÷ b₂ = c₁ ÷ c₂

Both equations represent the same line. The system is consistent but dependent.

Three unknowns strategy

Eliminate one variable from two pairs of equations, then solve the resulting 2 × 2 system

Pick the variable that is easiest to cancel. Find the third variable by back-substitution.

Standard form

ax² + bx + c = 0, a ≠ 0

Move every term to one side and arrange in powers of x before you read a, b, c.

Quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Works for every quadratic. The ± gives the two roots.

Discriminant

D = b² − 4ac

Only the sign of D and whether it is zero or a perfect square matter for nature of roots.

Nature of roots

D > 0: real and distinct; D = 0: real and equal; D < 0: not real

For rational a, b, c: D a positive perfect square means rational roots; D positive but not a perfect square means irrational roots.

Sum and product of roots

α + β = −b ÷ a; αβ = c ÷ a

Useful for checking your answers quickly.

Equation from roots

x² − (α + β)x + αβ = 0

Use when the roots are given and you need the equation.

Sum of roots

α + β = −b/a

For ax² + bx + c = 0 with a ≠ 0. Watch the minus sign.

Product of roots

αβ = c/a

No minus sign. Divide by a, the coefficient of x².

Equation from roots

x² − (α + β)x + αβ = 0

Multiply by a common denominator if the coefficients are fractions.

Sum of squares

α² + β² = (α + β)² − 2αβ

Most common expression in exams.

Difference of roots

(α − β)² = (α + β)² − 4αβ

Also (α − β)² = D/a², where D = b² − 4ac. Take the square root carefully for the sign.

Sum of cubes

α³ + β³ = (α + β)³ − 3αβ(α + β)

Also written (α + β)(α² − αβ + β²).

Sum of reciprocals

1/α + 1/β = (α + β) ÷ αβ

Valid only when αβ ≠ 0, that is c ≠ 0.

Ratio form

α/β + β/α = (α² + β²) ÷ αβ

Find α² + β² first.

Equation with reciprocal roots

cx² + bx + a = 0

Swap a and c. Needs c ≠ 0.

Equation with negative roots

ax² − bx + c = 0

Roots are −α and −β. Only the sign of b changes.

Equation with roots kα and kβ

ax² + kbx + k²c = 0

Sum becomes k times the old sum, product k² times the old product.

Standard form

ax² + bx + c = 0, a ≠ 0

Bring every term to one side before solving.

Quadratic formula

x = [−b ± √(b² − 4ac)] ÷ 2a

Works for every quadratic. Use it when factorising is not quick.

Sum and product of roots

α + β = −b ÷ a; αβ = c ÷ a

Useful for checking your roots quickly.

Consecutive numbers

x, x + 1, x + 2 (integers); x, x + 2, x + 4 (consecutive even or odd)

Choose the form that matches the wording.

Area of rectangle

Area = length × breadth; Perimeter = 2 (length + breadth)

Perimeter gives a sum, area gives a product.

Speed, distance, time

Time = Distance ÷ Speed

Equating a time difference creates the quadratic.

Pythagoras theorem

hypotenuse² = base² + perpendicular²

Used in right-angled triangle problems.

Factor theorem

(x − a) is a factor of f(x) ⇔ f(a) = 0

If f(a) = 0, then a is a root. If the remainder on dividing by (x − a) is not 0, then a is not a root.

Remainder theorem

Remainder when f(x) is divided by (x − a) = f(a)

Synthetic division gives this remainder as its last number.

Trial roots for integer coefficients

Any integer root divides the constant term. For a rational root p/q in lowest terms, p divides the constant term and q divides the leading coefficient.

Use this to build your list of values to test. A root need not be an integer if the leading coefficient is not 1.

Roots of ax³ + bx² + cx + d = 0 (a ≠ 0)

α + β + γ = −b ÷ a; αβ + βγ + γα = c ÷ a; αβγ = −d ÷ a

Use these to check your roots or to eliminate options quickly.

Quick root checks

If a + b + c + d = 0, then x = 1 is a root. If −a + b − c + d = 0, then x = −1 is a root.

These are just f(1) = 0 and f(−1) = 0 for a cubic.

Number of roots

A polynomial equation of degree n has at most n roots

A cubic with real coefficients always has at least one real root. The other two may be real or non-real.

Total cost (linear)

C = F + v·x

F = fixed cost, v = variable cost per unit, x = units produced.

Revenue

R = p·x

p = selling price per unit, x = units sold.

Profit

P = R − C

Profit is zero at break-even. A negative value means loss.

Break-even quantity (linear)

x = F ÷ (p − v)

Valid when p > v. The term (p − v) is the contribution per unit.

Break-even sales value

Sales value = p × F ÷ (p − v)

Break-even units multiplied by the price.

Average cost

AC = C ÷ x

Total cost divided by units. Do not confuse it with variable cost per unit.

Market equilibrium

Quantity demanded = Quantity supplied

Solve for price first. Then substitute to find quantity.

Quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Use when the break-even equation is ax² + bx + c = 0 and does not factorise easily.

Quick revision

• A linear equation in one variable has one solution: ax + b = 0 gives x = -b ÷ a, for a ≠ 0.
• In simultaneous equations, multiply to match one coefficient, then add or subtract to eliminate a variable.
• A quadratic ax² + bx + c = 0 needs a ≠ 0 and has at most two real roots.
• Roots by formula: x = (-b ± √(b² - 4ac)) ÷ 2a.
• Discriminant D = b² - 4ac. If D > 0, roots are real and distinct. If D = 0, roots are real and equal. If D < 0, no real roots.
• Sum of roots = -b ÷ a. Product of roots = c ÷ a.
• A quadratic with roots α and β is x² - (α + β)x + αβ = 0.
• If D is a perfect square and a, b, c are rational, the roots are rational.
• In word problems, reject roots that are negative or fractional when the quantity must be a positive whole number.
• For cubics and higher degrees, test small values like 1, -1, 2, -2 to find one root, then factorise.
• If x = k is a root, then (x - k) is a factor.
• Break-even occurs where total revenue equals total cost.

Common mistakes

• Not changing the sign when moving a term across the equals sign. Fix: Write the operation on both sides, for example '− 5 on both sides', for the first few practice questions. Then it becomes automatic.
• Multiplying only the first term inside a bracket, such as 2(x − 4) = 2x − 4. Fix: Multiply every term inside the bracket. 2(x − 4) = 2x − 8. Draw arrows from the outside number to each term.
• Sign errors when subtracting equations Fix: Write the second equation with all signs reversed on a new line, then add. Or always eliminate by adding after multiplying by a negative number.
• Multiplying only the left side when scaling an equation Fix: When you multiply an equation by k, multiply every term, including the constant on the right.
• Reading a, b, c before moving all terms to one side. Fix: Rewrite as x² − 5x + 6 = 0 first. Then a = 1, b = −5, c = 6.
• Dropping the sign of b in −b or in b². Fix: Write b in brackets. For b = −5, −b = 5 and b² = 25.
• Taking the sum of roots as b/a instead of −b/a. Fix: Say it as "sum is minus b over a, product is c over a". Test with x² − 5x + 6 = 0: roots 2 and 3, sum 5 = −(−5)/1.
• Forgetting to divide by a when a ≠ 1. Fix: Always write S = −b/a and P = c/a with the actual value of a. Here S = 3 and P = 3/2.
• Keeping both roots as the answer Fix: Reject negative lengths, speeds, ages and fractional counts before choosing the option.
• Giving x as the answer when the question asks for something else Fix: Underline what is asked, such as the larger number or the perimeter, and compute that last.

Exam tips

• In age problems, build a small table with the columns 'now' and 'after/before'. This avoids the commonest error of changing only one age.
• If the options are whole numbers and the equation is messy, substitute the options. Start with a middle value to save time.
• Always read the last line of the question again. Many options are traps that give x or a related value rather than the quantity asked.
• For equations with fractions, multiply by the LCM first. Do not add fractions one by one.
• Treat a question as a skip if the setup is not clear within 30 seconds. Negative marking of 0.25 per wrong answer makes blind guessing costly.
• Read the last line of the question first. Often you need x + y or a similar combination, which you can get by adding the equations.
• In MCQs, substitute options into the simplest equation to cut down choices quickly.
• For word problems, define the variables on paper in one line. Most errors come from setting up the wrong equation, not from solving it.