Quick Revision

Square Identities

• $(a \pm b)^2 = a^2 \pm 2ab + b^2$ and $(a + b)^2 - (a - b)^2 = 4ab$.
• With $s = a + b$, $p = ab$: $a^2 + b^2 = s^2 - 2p$, $(a - b)^2 = s^2 - 4p$.
• $\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}$; $a^4 + b^4 = (a^2 + b^2)^2 - 2p^2$.
• Rectangle: half-perimeter $s$, area $p$, diagonal $\sqrt{s^2 - 2p}$.
• $a - b = \pm\sqrt{s^2 - 4p}$, and the question decides the sign.

Cube Identities

$$(a + b)^3 = a^3 + b^3 + 3ab(a + b)$$

$$(a - b)^3 = a^3 - b^3 - 3ab(a - b)$$

• $a^3 + b^3 = (a + b)(a^2 - ab + b^2) = s^3 - 3ps$.
• $a^3 - b^3 = (a - b)(a^2 + ab + b^2) = d^3 + 3pd$, with $d = a - b$.
• Scaled letters: for $2x + 3y$, use $A = 2x$ and $B = 3y$, so $AB = 6xy$.
• In scaled questions the product is $AB$, not $xy$.

Factorising and Algebraic Fractions

• Order: common factor, grouping, an identity, then splitting the middle term.
• $x^3 - 3kx^2 + 3k^2x - k^3 = (x - k)^3$.
• $a^4 + a^2b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)$.
• $\frac{a^3 - b^3}{a^2 + ab + b^2} = a - b$ and $\frac{a^3 + b^3}{a^2 - ab + b^2} = a + b$.
• Test options with $a = 1$, $b = 0$, then $a = 2$, $b = 1$.
• When testing, never use values that make a bottom zero.

Three-Variable Identities

$$(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$$

$$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$$

• The second bracket is $\frac{1}{2}\left[(a - b)^2 + (b - c)^2 + (c - a)^2\right]$.
• $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{ab + bc + ca}{abc}$.
• Trap: the bracket takes away $ab + bc + ca$ once, not twice.

When $a + b + c = 0$

• $a^3 + b^3 + c^3 = 3abc$.
• $(x - y)^3 + (y - z)^3 + (z - x)^3 = 3(x - y)(y - z)(z - x)$.
• Shifted: $p + q + r = 15$ makes $(p - 3) + (q - 7) + (r - 5) = 0$.
• $a + b = c$ gives $a^3 + b^3 - c^3 = -3abc$.
• $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} = 3$ and $(a + b)(b + c)(c + a) = -abc$.
• Keep every minus sign when you write $3abc$.

Special Conditions

• Sum of squares $= 0$: each is 0.
• $a^2 + b^2 + c^2 = ab + bc + ca$, or $a^3 + b^3 + c^3 = 3abc$ with $a + b + c \neq 0$: then $a = b = c$.
• $8a^3 + b^3 + 27c^3 = 18abc$: if the values are positive, $2a = b = 3c$. Without that condition, either $2a = b = 3c$ or $2a + b + 3c = 0$.
• Trap: that rule works only because 18 is exactly 3 times $2 \times 1 \times 3$, the cube roots of the three coefficients multiplied. With any other number on the right, it does not apply.
• $a + b = 2c$ gives $(a - c)^3 + (b - c)^3 = 0$.

$x + \frac{1}{x}$ and $x - \frac{1}{x}$

With $k = x + \frac{1}{x}$ and $m = x - \frac{1}{x}$:

$$x^2 + \frac{1}{x^2} = k^2 - 2 = m^2 + 2$$

$$x^3 + \frac{1}{x^3} = k^3 - 3k \qquad x^3 - \frac{1}{x^3} = m^3 + 3m$$

• Fourth power $= \left(x^2 + \frac{1}{x^2}\right)^2 - 2$; sixth $= \left(x^3 + \frac{1}{x^3}\right)^2 - 2$.
• $k^2 - m^2 = 4$ and $x^2 - \frac{1}{x^2} = km$.
• $x^2 - kx + 1 = 0$: divide by $x$.
• $\left(3x + \frac{1}{2x}\right)^2$ has middle term 3, not 2.

Special Values and Cyclic Powers

Values of $x + \frac{1}{x}$:

• 2 gives $x = 1$, and $-2$ gives $x = -1$.
• 1, that is $x^2 - x + 1 = 0$: $x^3 = -1$ and $x^6 = 1$.
• $-1$, that is $x^2 + x + 1 = 0$: $x^3 = 1$.
• 0 gives $x^2 = -1$; $\sqrt{2}$ gives $x^4 = -1$; $\sqrt{3}$ gives $x^6 = -1$.
• Cut powers with the cycle: $x^3 = 1$ makes $x^{100} = x$.
• Trap: 1 gives $x^3 = -1$, not 1. Common in the PYQ Practice Sets.

Surd Values of $x$

• $x = a + \sqrt{b}$ with $a^2 - b = 1$: $\frac{1}{x} = a - \sqrt{b}$, $x + \frac{1}{x} = 2a$, $x - \frac{1}{x} = 2\sqrt{b}$.
• $x = \sqrt{a} + \sqrt{b}$ with $a - b = 1$: $x + \frac{1}{x} = 2\sqrt{a}$.
• $\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 = x + \frac{1}{x} + 2$.
• Otherwise use the conjugate $y$: $x + y = 2a$, $xy = a^2 - b$.
• Cut powers: $2 + \sqrt{3}$ solves $x^2 - 4x + 1 = 0$.
• If $x < 1$, then $x - \frac{1}{x}$ is negative.

Polynomials

• $f(x) \div (x - a)$ leaves remainder $f(a)$; $f(x) \div (ax - b)$ leaves $f\left(\frac{b}{a}\right)$.
• $x - a$ is a factor exactly when $f(a) = 0$. Use this to find $k$.
• Subtract $f(a)$ to make $x - a$ a factor.
• Spot cubes: $a^3 + 9a^2 + 27a = 37$ means $(a + 3)^3 = 64$.
• [Tier 2] For $ax^3 + bx^2 + cx + d$: zeros add up to $-\frac{b}{a}$, pair products to $\frac{c}{a}$, product $-\frac{d}{a}$.
• Dividing by $x + 2$ means putting $x = -2$.

Quadratic Equations

• $ax^2 + bx + c = 0$ has roots $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Sum of roots $= -\frac{b}{a}$ and product $= \frac{c}{a}$.
• $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ and $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$.
• Equation from its roots: $x^2 - (\text{sum})x + \text{product} = 0$.
• $b^2 - 4ac$ positive: two real roots; zero: equal roots; negative: no real roots.
• One root known: the other is the product divided by it.
• Trap: the sum is $-\frac{b}{a}$, not $\frac{b}{a}$.

Linear Equations

• Two letters: substitution or elimination.
• Swapped coefficients: add the equations to get $x + y$, subtract to get $x - y$.
• Three equations, each missing one letter: add all three and divide.
• One solution if $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$; none if $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$; infinitely many if all three ratios are equal.
• Fractions in an equation: multiply through by the LCM of the bottoms first.
• Trap: in a no-solution question, reject the $k$ that makes all three ratios equal.

Word Problems

• A two-digit number is $10a + b$; with its digits swapped it is $10b + a$.
• Their difference is $9(a - b)$ and their sum is $11(a + b)$.
• Fractions: write $\frac{n}{d}$, change the top and bottom as told, then cross-multiply.
• Prices of items: one letter per item. For an asked mix, add or combine the given lines.
• Tickets or notes: one equation for the count, one for the money.
• Digits must be whole numbers from 0 to 9.

Maximum and Minimum Values

• $x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 + c - \frac{b^2}{4}$, least at $x = -\frac{b}{2}$.
• [Tier 2] $ax^2 + bx + c$ has least value ($a > 0$) or greatest value ($a < 0$) $c - \frac{b^2}{4a}$.
• For $x > 0$: $x + \frac{1}{x} \ge 2$ and $ax + \frac{b}{x} \ge 2\sqrt{ab}$.
• Fixed sum: the product is greatest when the parts are equal.
• Read what is asked: it may be the $x$, not the least value.