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Maths

Example output

Methods and worked steps, with the reasoning made explicit.

Source: maths-quadratics.jpg · handwritten

Summary

Your working on quadratics is correct throughout, but several solutions skip the step where the discriminant is evaluated — you get the right roots, yet an exam wants that line shown. The completing-the-square method is missing entirely.

Solving quadratics

Factorising and the formula are both used correctly; completing the square doesn't appear.

The discriminant

Used implicitly to decide the number of roots, but never written out as a step.

Key terms

Discriminantpage 1
b² − 4ac — determines how many real roots a quadratic has.
Rootpage 1
A value of x for which the expression equals zero.
Completing the squarepage 2
Rewriting ax² + bx + c in the form a(x + p)² + q.
Formula sheet built from your working
x = (−b ± √(b² − 4ac)) / 2a
the quadratic formula
b² − 4ac > 0
two distinct real roots
b² − 4ac = 0
one repeated root
b² − 4ac < 0
no real roots

Flashcards

What does b² − 4ac < 0 mean?
No real roots — the parabola never crosses the x-axis.
State the quadratic formula
x = (−b ± √(b² − 4ac)) / 2a
Why complete the square?
It reveals the turning point directly and solves any quadratic.

Practice question

How many real roots does x² + 2x + 5 = 0 have?

Pick an answer to see the reasoning.

Study plan

  1. 1
    Learn completing the square from scratch
    25 min
  2. 2
    Show the discriminant explicitly in 6 problems
    20 min
  3. 3
    Mixed quadratics past paper
    20 min

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