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Quadratics by inspection

Roots of x² − 5x + 6 at a glance: two numbers with the right sum and product. Then leading coefficients, the discriminant, and sum and product of roots.

  • Quantitative reasoning
  • Untimed
  • Endless levels

How it plays

x² − 5x + 6 = 0 has roots 2 and 3, because 2 + 3 = 5 and 2 × 3 = 6. Every root question here is that search: the sum is minus the x coefficient, the product is the constant.

Levels bring negative roots, a leading coefficient to factor out (6x² − x − 2 = (3x − 2)(2x + 1)), the nature of the roots from b² − 4ac, and the sum −b/a and product c/a of the roots without solving at all. Roots stop growing at about ±30, where the formula becomes the faster route.

Every equation is built from its roots, so the answer is known, not searched for. The wrong options are the real slips: both signs flipped, a factor pair with the right product but the wrong sum, the leading coefficient forgotten.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes solving and reading quadratics by inspection quicker and surer. It claims nothing wider.

Learn

How to get better at quadratics by inspection: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Read the question first: the roots, the kind of roots, or their sum or product. Only the first needs you to solve anything.
  2. For x² + bx + c, write down the two targets: the roots add to −b and multiply to c. In x² − 5x + 6 they add to 5 and multiply to 6.
  3. Settle the signs before the sizes. A positive c means both roots have the same sign, the sign of −b; a negative c means one of each.
  4. Then search the factor pairs of c for the one with the right sum, and check by putting a root back in. With a leading coefficient, divide it out of the targets or factorise by grouping.

Techniques and tricks

  • Sum and product

    If the roots are r and s, then x² + bx + c = (x − r)(x − s), so r + s = −b and r × s = c. Every root question is the search for two numbers with that sum and that product.

    e.g. x² − 14x + 48: sum 14, product 48 → 6 and 8

  • Signs first, from c

    If c is positive the roots share a sign, the same as −b’s. If c is negative they differ, and the larger one in size takes −b’s sign. Then look only for sizes.

    e.g. x² + 2x − 15: product −15, so one of each; sum −2, so the larger is negative: −5 and 3

  • Walk the factor pairs in order

    List the factor pairs of |c| from 1 × c inwards. For a positive c you want the pair that adds to the sum; for a negative c, the pair that differs by it. The pairs get closer as you go, so you know which way to walk.

    e.g. 48: 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8 — sum 14 is the last

  • A leading coefficient: split the middle

    For ax² + bx + c, find two numbers that multiply to a × c and add to b. Split the middle term with them and factor in pairs. A root of (ax − p) is p ÷ a, not p.

    e.g. 6x² − x − 2: a × c = −12, pair −4 and 3: 6x² − 4x + 3x − 2 = (3x − 2)(2x + 1), roots 2/3 and −1/2

  • The discriminant without the formula

    Work out b² − 4ac: positive means two different real roots, zero one repeated root, negative none. If a and c have opposite signs, −4ac is positive and there are always two roots.

    e.g. 2x² + 3x + 5: 9 − 40 = −31, no real roots; x² − 6x + 9: 36 − 36 = 0, one repeated root

  • Sum −b/a, product c/a

    With a leading coefficient, divide through by a: the roots add to −b/a and multiply to c/a. You never need the roots themselves to answer these.

    e.g. 4x² − 6x + 9: sum 6/4 = 3/2, product 9/4

As the levels climb

  • The kinds unlock in this order: positive roots; then roots of either sign; then a leading coefficient; then the nature of the roots; then the sum and product of the roots. Four in ten questions come from the newest kind.
  • The roots grow as you climb, from up to 6 to as far as ±30, and then stop: past that the formula is quicker than inspection. With large constants, walk the factor pairs systematically rather than guessing.
  • With a leading coefficient, expect a fraction in one root. If your factors give two whole roots, check whether the coefficient was divided.
  • Each step of size adds 8% to the time, up to twelve steps, so the larger roots also get longer to find.

Traps to avoid

  • Both signs flipped: x = −2 and x = −3 for x² − 5x + 6. The factors are (x − 2)(x − 3), which are zero at +2 and +3. The roots add to −b, not b.
  • A factor pair with the right product but the wrong sum: 1 and 6 for x² − 5x + 6. Always check the sum as well as the product.
  • The leading coefficient forgotten: 7 and −5 for 4x² + 13x − 35. The bracket (4x − 7) is zero at x = 7/4. The upside-down 4/7 is the same slip in another form.
  • One sign flipped: −3 and 5 where 3 and −5 are right. With a negative c, check which root takes −b’s sign by adding them.
  • The sign of b kept in the sum of roots: b/a instead of −b/a. Or −c/a for the product. Match the coefficients of a(x − r)(x − s) once and you will not swap them again.

How to practise

  • In “Kinds of question”, drill “Roots of either sign” on its own until the sign rule from c is instant; it halves the search.
  • Switch “Answer by” to “Ticking every right one” for the root questions. You must pick each root from a list that includes the negatives and the other factors, not choose a ready-made pair.
  • Raise “Options shown” towards 8 so the flipped, the wrong-pair and the coefficient-forgotten answers sit side by side.
  • Read the working on each miss: it shows the factorised form and checks the middle term, which is the check to run yourself.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More quantitative reasoning

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