Special Manipulations: Chapter Review

Vocabulary and notation

Equivalent equations

Two equations with the same solution set. Every reversible step gives one; an even power need not.

Radical equation

An equation hiding the variable under a root, such as $\sqrt{2x - 1} = 5$.

Extraneous solution

A candidate that solves the raised or cleared equation but fails the original, so it is discarded.

Principal square root

The nonnegative value $\sqrt{\phantom{x}}$ returns, so $\sqrt{9} = 3$ and never $-3$. An odd root is unrestricted: $\sqrt[3]{-8} = -2$.

Self-similar expression

An infinite expression containing a perfect copy of itself, so naming the whole $x$ makes the inner copy $x$ too.

Continued fraction

A stack of divisions running on forever, such as $1 + \frac{1}{1 + \frac{1}{1 + \cdots}}$.

Golden ratio $\varphi$

The positive number equal to one plus its own reciprocal, $\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618$.

Symmetric expression

An expression in $x$ and $y$ unchanged by swapping the letters. Both $x + y$ and $xy$ pass, and so does $x^2 + y^2$; $x - y$ and $x^2 y$ fail.

The sum $s$ and the product $p$

The abbreviations $s = x + y$ and $p = xy$, the two quantities blind to the swap that every symmetric polynomial is built from.

Palindromic (reciprocal) equation

A polynomial equation whose coefficient list reads the same forwards and backwards, such as $2x^4 - 9x^3 + 14x^2 - 9x + 2 = 0$.

Formulas and theorems

• Raising both sides to a power

  An odd power gives an equivalent equation. An even power keeps every solution and can add candidates the original never had.

  Use when Real values. An even power hides the sign of its base, so $a^n = b^n$ leaves open $a = b$ or $a = -b$; an odd power does not. After any even power, check every candidate in the original.

• Square root of a square

  $$\sqrt{x^2} = \lvert x \rvert$$

  Use when Every real $x$. Order matters: rooting first gives $\left(\sqrt{x}\right)^2 = x$ exactly, but that needs $x \ge 0$.

  e.g. $\sqrt{(-7)^2} = 7$, not $-7$.

• Taking an even root of both sides

  $$\begin{gathered} (x - h)^2 = k \\ \Longrightarrow\ \lvert x - h \rvert = \sqrt{k} \\ \Longrightarrow\ x - h = \pm\sqrt{k} \end{gathered}$$

  Use when $k \ge 0$ for real solutions. The absolute value forces the two signs, and both are genuine: rooting undoes a square rather than performing one, so nothing extraneous appears.

  e.g. $(x + 2)^2 = 9$ gives $x = 1$ or $x = -5$.

• Multiplying or dividing by an expression that can vanish

  Multiplying both sides by an expression that can be zero adds solutions; dividing by one loses them.

  Use when Whenever the multiplier or divisor holds the variable. After multiplying, test every candidate in the ORIGINAL and discard any it leaves undefined. Never divide by a variable expression: factor instead, since $x^2 = 5x$ divided by $x$ drops $x = 0$.

• Self-similar towers

  $$\begin{gathered} x = \sqrt{c + x} \ \Longrightarrow \ x^2 - x - c = 0 \\ x = \sqrt{cx} \ \Longrightarrow \ x(x - c) = 0 \end{gathered}$$

  Use when $c > 0$. Every layer is strictly positive, so keep the strictly positive root, not merely a nonnegative one: the additive tower has only one nonnegative root, while $x(x - c) = 0$ offers two, and the value there is $x = c$, never $0$.

• Continued fraction

  $$x = a + \frac{1}{x} \quad\Longrightarrow\quad x^2 - ax - 1 = 0$$

  Use when $a > 0$ and $x \neq 0$. The parts are all positive, so keep the positive root. With $a = 1$ this is $\varphi$.

  e.g. $a = 2$ gives $x = 1 + \sqrt{2} \approx 2.414$, rejecting $1 - \sqrt{2}$.

• Repeating decimal as a shifted copy

  $$10^k x = B + x \quad\Longrightarrow\quad x = \frac{B}{10^k - 1}$$

  Use when $x = 0.\overline{B}$ with a $k$-digit block starting right after the decimal point. The power of ten must match the block length, or the tail will not line up. Convergence is never in question here: a decimal already names a point on the number line.

  e.g. $x = 0.\overline{27}$: $100x = 27 + x$, $99x = 27$, $x = \tfrac{3}{11}$.

• Infinite geometric series, self-similar form

  $$S = a_1 + rS \quad\Longrightarrow\quad S = \frac{a_1}{1 - r}$$

  Use when $\lvert r \rvert < 1$. Outside that range the series has no sum, so whatever the formula returns means nothing.

• What the self-similar equation proves

  It gives the value the expression must have IF it settles on a number, and never proves that it does.

  Use when Settle that separately first: $\lvert r \rvert < 1$ for a geometric series, or partial values that climb while held under a fixed ceiling for a tower.

• Sum and product reductions

  $$\begin{gathered} x^2 + y^2 = s^2 - 2p \\ x^3 + y^3 = s^3 - 3ps \\ \frac{1}{x} + \frac{1}{y} = \frac{s}{p} \\ (x - y)^2 = s^2 - 4p \end{gathered}$$

  Use when $s = x + y$, $p = xy$; every symmetric polynomial in $x$ and $y$ reduces this way. The reciprocal form needs $x \neq 0$ and $y \neq 0$. Reapply $x^2 + y^2 = s^2 - 2p$ to $x^2$ and $y^2$ for $x^4 + y^4 = (s^2 - 2p)^2 - 2p^2$.

• A symmetric system as one quadratic

  $$t^2 - st + p = 0$$

  Use when Monic, minus the sum and plus the product. Its roots are the unordered pair $\{x, y\}$, so the method never says which is which. The pair is real exactly when $s^2 \ge 4p$, since $(x - y)^2 = s^2 - 4p$, and a complex conjugate pair otherwise.

• Pairing a number with its reciprocal

  $$\begin{gathered} k = x + \frac{1}{x} \\ x^2 + \frac{1}{x^2} = k^2 - 2 \\ x^3 + \frac{1}{x^3} = k^3 - 3k \end{gathered}$$

  Use when $x \neq 0$. These are the $s$ and $p$ identities with the product fixed at $p = 1$. A real $x$ forces $\lvert k \rvert \ge 2$, since $k^2 - 4 = \left(x - \tfrac{1}{x}\right)^2$.

  e.g. $k = 4$: $x^2 + \tfrac{1}{x^2} = 14$ and $x^3 + \tfrac{1}{x^3} = 52$.

• Axis of symmetry

  $$x = -\frac{b}{2a}$$

  Use when $a \neq 0$, for $ax^2 + bx + c = 0$. The roots differ only by the sign before $\sqrt{b^2 - 4ac}$, so they always average to this line, and when $b^2 - 4ac \ge 0$ they sit the same distance either side of it; the vertex rides on it.

  e.g. $x^2 - 6x + 5 = 0$ has roots $1$ and $5$, mirrored about $x = 3$.

Problem types, step by step

Exam traps

• Trap Reporting every candidate the algebra survives: squaring $\sqrt{x + 2} = x$ gives $x^2 - x - 2 = 0$, answered as $x = 2$ and $x = -1$.

  Fix Squaring is not reversible, so test both in the ORIGINAL. At $x = 2$: $\sqrt{4} = 2$, true. At $x = -1$: $\sqrt{1} = 1 \neq -1$, so it is extraneous. The solution is $x = 2$ alone.

• Trap Squaring term by term, turning $\sqrt{x} = x - 2$ into $x = x^2 + 4$.

  Fix The right side is a binomial and must be expanded in full: $x = x^2 - 4x + 4$. For the same reason $\sqrt{a + b}$ is not $\sqrt{a} + \sqrt{b}$.

• Trap Assuming the negative candidate, or the smaller one, is automatically the extraneous one.

  Fix Only substitution decides. In $\sqrt{3x + 1} - \sqrt{x + 4} = 1$ the smaller candidate $x = 0$ fails, while in $x^{2/3} = 4$ both $8$ and $-8$ are genuine.

• Trap Trusting a self-similar value without asking whether the expression settles: $1 + 2 + 4 + 8 + \cdots$ gives $S = 1 + 2S$, hence $S = -1$.

  Fix Every line of that algebra is right; writing $S =$ was not. The running totals $1, 3, 7, 15, \ldots$ pass every bound, so there is no number to name. Grandi's $1 - 1 + 1 - \cdots$ fails the same way: bracketing gives $0$ or $1$, and the self-similar move gives $\tfrac{1}{2}$.

• Trap Misreading the copy inside, turning $\sqrt{2 + \sqrt{2 + \cdots}}$ into $x = 2 + \sqrt{x}$ or $x = \sqrt{2} + x$.

  Fix The copy must be the whole expression again. Under the outermost radical sits $2$ plus another full tower, so the equation is $x = \sqrt{2 + x}$.

• Trap Slipping a sign or a factor in the reductions: $x^2 + y^2 = s^2 + 2p$, or $x^3 + y^3 = s^3 - 3p$.

  Fix The cross term is subtracted, and the cubic correction carries a factor of $s$. With $s = 6$, $p = 4$: $x^2 + y^2 = 36 - 8 = 28$, not $44$; and $x^3 + y^3 = 216 - 72 = 144$, not $204$.