Equations and Inequalities — IMAT Mathematics Formulas and Traps

Expanding, Factorising and Algebraic Fractions

Expanding brackets and the special products

Special products

(a \pm b)^2 = a^2 \pm 2ab + b^2 \qquad (a+b)(a-b) = a^2 - b^2

Factorising: common factors, grouping and trinomials

Monic trinomial

x^2 + sx + p = (x + m)(x + n) \quad\text{where}\quad m + n = s,\; mn = p

Simplifying and combining algebraic fractions

Adding fractions

\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}

Polynomial division and the factor theorem

Remainder theorem

p(x) = (x - a)\,q(x) + p(a)

$q(x)$ the quotient, one degree lower than $p(x)$
$p(a)$ the remainder, a number

Common traps

The square of a sum is not the sum of the squares

(a + b)^2 = a^2 + 2ab + b^2

, not

a^2 + b^2

. Dropping the middle term

2ab

is the commonest expansion error, and IMAT offers the result as an option.

A sum of two squares does not factorise

x^2 - 9 = (x - 3)(x + 3)

, but

x^2 + 9

has no real factors:

(x + 3)(x - 3)

expands to

x^2 - 9

, and

(x + 3)^2

has a middle term.

Cancel factors, never terms

\frac{x + 5}{x + 7}

is not

\frac{5}{7}

: the

x

is a term added on, not a factor of the whole top and bottom. Factorise first, then cancel only whole brackets that multiply everything.

For the factor x + a, test x = minus a

x + a

is zero when

x = -a

, so that is the value to substitute. Substituting

x = a

answers a different question, and its result is always among the options.

Linear Equations, Parameters and Systems

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Linear equations and equations with a parameter

Solution of ax = b

ax = b,\; a \ne 0 \;\Rightarrow\; x = \frac{b}{a}

Systems of two linear equations

Two equations in two unknowns

\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}

Common traps

Zero coefficient: check the right-hand side

When the coefficient of

x

is zero, the equation is not automatically impossible. If the right-hand side is also zero, every

x

works. Only a zero coefficient with a non-zero right-hand side gives no solution.

Proportional coefficients do not always mean no solution

If the

x

and

y

coefficients are in proportion, the lines are parallel. They have no common point only if the constants break the proportion; if the constants follow it too, the lines coincide and there are infinitely many solutions.

Quadratic Equations, the Discriminant and the Roots

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Solving quadratic equations and the discriminant

Quadratic formula

x = \frac{-b \pm \sqrt{\Delta}}{2a}, \qquad \Delta = b^2 - 4ac

$a, b, c$ the coefficients of $x^2$ , $x$ and the constant
$\Delta$ the discriminant

Sum and product of the roots

x_1 + x_2 = -\frac{b}{a} \qquad x_1 x_2 = \frac{c}{a}

Common traps

Never divide both sides by x

Dividing

3x^2 = 12x

by

x

gives only

x = 4

and loses

x = 0

. Bring everything to one side and factorise out

x

instead.

The sum of the roots is minus b over a

Expanding

a(x - x_1)(x - x_2)

gives

-a(x_1 + x_2)

as the

x

coefficient, so the sum is

-\frac{b}{a}

, not

\frac{b}{a}

. And check

\Delta \ge 0

first:

x^2 + 2x + 5 = 0

has no real roots, so it has no real roots to add.

Linear, Quadratic, Fractional and Absolute Value Inequalities

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Linear inequalities and systems of inequalities

Multiplying by a negative number

a < b \;\text{and}\; c < 0 \;\Rightarrow\; ac > bc

Quadratic inequalities with a sign chart

Sign of a quadratic with a > 0 and roots x₁ < x₂

a(x - x_1)(x - x_2) < 0 \;\text{for}\; x_1 < x < x_2, \qquad > 0 \;\text{for}\; x < x_1 \;\text{or}\; x > x_2

Inequalities with absolute value

Absolute value inequalities (k > 0)

|A| < k \;\Leftrightarrow\; -k < A < k \qquad |A| > k \;\Leftrightarrow\; A < -k \;\text{or}\; A > k

Common traps

Dividing by a negative reverses the sign

From

-3x < 6

the answer is

x > -2

, not

x < -2

. The unreversed answer is always among the options.

x² < 16 is not x < 4

Taking the square root of both sides loses the negative half:

x = -10

satisfies

x < 4

but not

x^2 < 16

. Write

(x - 4)(x + 4) < 0

and read the answer

-4 < x < 4

from the sign chart.

Do not multiply by a denominator of unknown sign

Multiplying

\frac{3}{x - 1} < 1

by

x - 1

assumes

x - 1 > 0

and loses every solution with

x < 1

. Bring everything to one side and use a sign chart.

Greater than gives two pieces, not one

|x + 2| > 3

is

x < -5

or

x > 1

. Writing it as

-5 > x > 1

describes no number at all. Only the less-than form gives a single interval.

Equations with Fractions, Roots, Powers and Logs

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Equations with the unknown in a denominator

Clearing the denominator

\frac{P(x)}{Q(x)} = R(x) \;\Rightarrow\; P(x) = R(x)\,Q(x), \quad Q(x) \ne 0

Equations and inequalities with square roots

Squaring a root equation

\sqrt{A} = B \;\Rightarrow\; A = B^2 \quad\text{(keep only solutions with}\; B \ge 0)

Exponential and logarithmic equations

Matching exponents or logs

a^{f(x)} = a^{g(x)} \;\Rightarrow\; f(x) = g(x) \qquad \log_a f(x) = \log_a g(x) \;\Rightarrow\; f(x) = g(x) > 0

Common traps

An excluded value can appear as an answer

After clearing denominators, the algebra may produce a value that makes an original denominator zero. It is not a solution. IMAT offers it as an option; the correct choice may be "no solution".

Squaring can add false solutions

Every solution of

\sqrt{A} = B

solves

A = B^2

, but not the other way round. Substitute each answer back into the original equation; any that makes the right-hand side negative is false.

A root inequality has a domain

\sqrt{x} < 3

is

0 \le x < 9

, not

x < 9

: negative

x

has no square root at all. Options that forget the lower limit are common.

Check the domain of every logarithm

Combining logs can produce an answer that makes an original argument zero or negative, such as

x = -5

in

\log x

. Such an answer is not a solution. Likewise, in a substitution

t = a^x

, a negative

t

gives no

x

.

Equations and Inequalities formulas

Expanding, Factorising and Algebraic Fractions

Expanding brackets and the special products

Factorising: common factors, grouping and trinomials

Simplifying and combining algebraic fractions

Polynomial division and the factor theorem

Common traps

Linear Equations, Parameters and Systems

Linear equations and equations with a parameter

Systems of two linear equations

Common traps

Quadratic Equations, the Discriminant and the Roots

Solving quadratic equations and the discriminant

Common traps

Linear, Quadratic, Fractional and Absolute Value Inequalities

Linear inequalities and systems of inequalities

Quadratic inequalities with a sign chart

Inequalities with absolute value

Common traps

Equations with Fractions, Roots, Powers and Logs

Equations with the unknown in a denominator

Equations and inequalities with square roots

Exponential and logarithmic equations

Common traps

More IMAT Mathematics formula sheets