The chapter on linear equations gave one rule: do the same thing to both sides. For harder equations, with fractions, square roots and powers, you need more moves, and some of them are traps. Squaring both sides can create solutions that are not real. Dividing by x can throw a real one away. This chapter collects the moves and tricks used to handle equations, says which ones are always safe, and shows how to use the risky ones without being caught.
By the end you can
- Tell a move that keeps the solutions from one that can add fake ones or lose real ones.
- Clear fractions by cross-multiplying, after noting which values the denominators forbid.
- Solve a square-root equation by isolating, squaring and checking.
- Spot a substitution, and factor instead of dividing by the unknown.
Solving an equation is a chain of equations, each one rewritten from the one before. What matters is how two neighbours in the chain are connected. A two-way arrow, ⇔, means the two equations are true for exactly the same x: the move can be undone, so no solution is gained or lost. A one-way arrow, ⇒, means every solution of the first is a solution of the second, but not necessarily the other way round: the second equation may have extra solutions that the first does not. Some moves even point the wrong way, so that solutions of the first go missing.
A one-way move is not forbidden. It is often the only way forward. It just means the final list of answers is a list of candidates, and each candidate must be tested in the original equation. A candidate that fails is called extraneous, or a fake solution: the moves created it, the original equation never had it. The opposite danger, a lost solution, is worse, because no check at the end can bring it back. The table lists the usual moves.
| Move | Risk | How to stay safe |
|---|---|---|
| Add or subtract the same thing on both sides | none | always safe |
| Multiply or divide both sides by a number | none, unless the number is 0 | never multiply or divide by 0 |
| Multiply both sides by an expression with x, such as x − 2 | can add fake solutions where it is 0 | write down where it is 0 first, and reject those values |
| Divide both sides by an expression with x | can lose the solutions where it is 0 | factor it out instead, then use the zero-product rule |
| Square both sides | can add fake solutions (A = −B) | check every candidate in the original, or keep the sign condition |
| Take the square root of both sides | can lose the negative solution | write ±: x² = 9 gives x = ±3 |
| Take the logarithm of both sides | none, if both sides are positive | a log needs a positive input: check the domain |
| Use a log rule, such as ln a + ln b = ln(ab) | can add fake solutions (ab > 0 also when a, b < 0) | check every candidate in the original |
Blue is the left side of the equation, amber the right side; a solution is an x where they meet. Every candidate is tested in the first equation: green is real, red is fake (a move created it), a dashed amber ring is a real solution a move lost. Before each move, predict which of the three it will be.
When each side is a single fraction, a/b = c/d, you can multiply both sides by b·d. On the left the b cancels, on the right the d cancels, and what is left is a·d = b·c. Drawn on the page, each numerator is multiplied by the opposite denominator, crosswise, hence the name. The move multiplies by b·d, an expression that may contain x, so it is safe only where b and d are not zero. Write those forbidden values down before you start.
x³ = 4x looks like an invitation to divide both sides by x. Do that and you get x² = 4, so x = ±2, and the solution x = 0 is gone: 0³ = 4 · 0 is true. Dividing by x assumed x ≠ 0 without saying so. The safe route moves everything to one side and factors out what you wanted to divide by. Then the zero-product rule finishes the job: a product is zero exactly when at least one of its factors is zero.
To get rid of a square root, square it: (√u)² = u. Two habits make this work. First, isolate the root, so it is alone on one side. If you square √(x + 7) + 5 as it is, (a + b)² = a² + 2ab + b² gives a middle term 10√(x + 7), and the root is still there. Second, squaring is a one-way move, so check every candidate in the original equation. A square root is never negative, so any candidate that makes the other side negative is fake.
A square root in a denominator, or a difference of two roots such as √(x + 5) − √x = 1, has its own trick: multiply by the conjugate, the same two terms with the sign between them flipped, because (a + b)(a − b) = a² − b² leaves no root behind. In an equation that means multiplying both sides, so the factor must not be zero, as in the table above. The trick has its own full section, with why it works, a picture and worked examples, in the chapter Algebra Tricks I at the end of this section.
Some equations are an old friend in disguise. x⁴ − 13x² + 36 = 0 has degree 4, but x appears only as x² and (x²)². Give the repeated piece a new name, u = x², and it becomes the quadratic u² − 13u + 36 = 0. Solve that, then undo the substitution. The same trick turns 4ˣ − 6 · 2ˣ + 8 = 0 into a quadratic, because 4ˣ = (2²)ˣ = (2ˣ)². Before undoing, look at what u is allowed to be: u = x² and u = 2ˣ can never be negative, so a negative u gives no x at all.
| When you see | Try | Watch out for |
|---|---|---|
| fractions with numbers below | multiply every term by the LCM (Linear Equations) | multiply every term, not just the fractions |
| one fraction on each side | cross-multiply | values that make a denominator 0 |
| the unknown as a common factor | move all to one side, factor, zero-product rule | never divide by it |
| x² and x, no way to factor | complete the square or the quadratic formula (Quadratics) | add the same number to both sides |
| a square root | isolate it, then square | fake solutions: check them all |
| a root in a denominator or a difference of roots | multiply by the conjugate | multiply top and bottom, or both sides |
| a square on each side, x² = 9 | square root with ±, or factor | the negative solution |
| the same block repeated (x², 2ˣ, sin x) | substitute u for it | values u cannot take |
| the unknown in an exponent | logarithm of both sides (Exponents & Logarithms) | a log needs a positive input |
| Wrong | Right | Why |
|---|---|---|
| √(a + b) = √a + √b | leave it as √(a + b) | √(9 + 16) = √25 = 5, but √9 + √16 = 7 |
| (x + 3)² = x² + 9 | x² + 6x + 9 | the cross term 2 · 3x is missing |
| 1/(√a + √b) = 1/√a + 1/√b | multiply by the conjugate | a fraction does not split over a sum in its denominator: 1/(1 + 1) = 1/2, not 2 |
| x² = 5x → x = 5 | x(x − 5) = 0: x = 0 or 5 | dividing by x lost x = 0 |
| x² = 16 → x = 4 | x = ±4 | (−4)² = 16 as well |
| squaring and not checking | test every candidate in the original | squaring can add fake solutions |
| x/2 + 1 = 3/x → x · x = 2 · 3 | multiply every term by 2x | cross-multiplying needs a single fraction on each side |
A round of 10 questions
Answer as many as you can. Mistakes come back at the end for a review round.
Every exercise is made on the spot, so a new round is never the same. The computer picks the answer first and builds the question around it, working with exact fractions, so the answer key is always right. Answers are read exactly: 0.75, 3/4 and 6/8 count as the same number (where lowest terms are asked for, 6/8 just gets a reminder). A wrong answer shows the worked solution line by line, in the order the lesson does it; the new number in each line is coloured.
Key ideas
- A move that can be undone (⇔) keeps the solutions; a one-way move (⇒) turns the answers into candidates that must be checked.
- Multiplying by an expression that can be 0, squaring, and some log rules can add fake solutions: check every candidate in the original.
- Dividing by an expression that can be 0, and taking a square root without ±, can lose real solutions: factor instead, and write ±.
- a/b = c/d ⇔ ad = bc, once b ≠ 0 and d ≠ 0 are written down; it needs one fraction on each side.
- For a square root: isolate it, square, then check.
- Substitution, u = x² or u = 2ˣ, turns a disguised equation into a familiar one; mind the values u cannot take.