Lesson 5 of 12 · Algebra Foundations
Quadratic Equations
Three methods solve every quadratic — factoring, completing the square, and the quadratic formula — but knowing which one is fastest for a given equation saves real time. The discriminant, meanwhile, often answers a question about the roots without requiring you to find them at all.
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Learning objectives
- Solve quadratics by factoring, completing the square, or the quadratic formula, and choose the fastest method on sight.
- Use the discriminant to determine the number and nature of roots without solving the equation.
- Convert a quadratic to vertex form to find its minimum or maximum value directly.
- Set up and solve a quadratic equation from a word problem, and reject solutions that don't fit the problem's constraints.
The core idea
Every quadratic ax² + bx + c = 0 can be solved by the quadratic formula, so that method always works — but it's rarely the fastest. If the quadratic factors over the integers, factoring is quicker and less error-prone. Completing the square is the right choice specifically when you need more than just the roots — the vertex, an extremum, or a proof about all real values of the expression. The discriminant is the fastest of all when a problem only asks about the roots' existence or type.
The quadratic formula
x = [−b ± √(b² − 4ac)] / (2a)
Always available, always correct — use it when factoring isn't obviously quick.
The discriminant
Δ = b² − 4ac
Δ > 0: two distinct real roots. Δ = 0: one repeated real root. Δ < 0: no real roots (two complex roots). You never need to compute the roots themselves to answer "how many real roots" or "is this a perfect square trinomial" — the discriminant alone answers both.
Vertex form
ax² + bx + c = a(x − h)² + k, where h = −b/(2a) and k = c − b²/(4a)
If a > 0, the minimum value is k, achieved at x = h. If a < 0, k is the maximum instead. This is the fastest route to any "find the minimum/ maximum value of..." question involving a quadratic.
Worked example 1 — Solving by factoring
Problem
Solve x² − 5x + 6 = 0.
Key insight
Two integers that multiply to 6 and add to −5: −2 and −3.
Solution
(x − 2)(x − 3) = 0, so x = 2 or x = 3.
Takeaway
Always check for integer factoring before reaching for the quadratic formula — it's faster whenever it works.
Worked example 2 — Discriminant without solving
Problem
Without solving, determine the number of real roots of 2x² − 4x + 5 = 0.
Key insight
Only the discriminant's sign is needed — compute it and stop.
Solution
Δ = (−4)² − 4(2)(5) = 16 − 40 = −24 < 0, so there are no real roots.
Takeaway
Never apply the quadratic formula just to answer a yes/no question about roots — the discriminant alone is the entire answer.
Worked example 3 — Completing the square for a minimum
Problem
Find the minimum value of f(x) = x² − 6x + 11.
Key insight
Complete the square: half of −6 is −3, and (−3)² = 9.
Solution
x² − 6x + 11 = (x − 3)² + 2. Since (x−3)² ≥ 0 always, the minimum value is 2, achieved at x = 3.
Takeaway
Completing the square answers "what's the minimum/maximum" directly — solving for roots wouldn't help here at all.
Worked example 4 — Word problem, with a rejected root
Problem
A rectangle's length is 3 more than its width, and its area is 40. Find its dimensions.
Key insight
Let w = width; then length = w + 3, and area gives a quadratic in w.
Solution
w(w + 3) = 40 → w² + 3w − 40 = 0 → (w + 8)(w − 5) = 0 → w = −8 or w = 5. A width can't be negative, so w = 5 and length = 8. The rectangle is 5 by 8.
Takeaway
A quadratic word problem almost always produces two mathematical roots but only one that satisfies the problem's real-world constraints — always check both.
Strategy notes
Try factoring first, but don't force it
A quick mental check (do two integers multiply to c and add to b?) costs almost nothing. If it doesn't work within a few seconds, move to the quadratic formula rather than searching indefinitely.
Match the method to the question
"Solve for x" → factoring or the quadratic formula. "How many real roots" → the discriminant alone. "Find the min/max value" → complete the square. Picking the method that directly answers the question skips unnecessary work.
Common mistakes
Sign errors in the quadratic formula
The formula is x = (−b ± √(b²−4ac)) / (2a) — the negative sign in front of b applies even when b itself is negative. Substitute carefully, keeping every sign explicit rather than simplifying mentally.
Forgetting to check both roots against the problem's constraints
As in Example 4, a mathematically valid root can be physically or logically impossible (negative length, non-integer count of people, etc.). Always check both roots before reporting the final answer.
Completing the square incorrectly when a ≠ 1
Factor out the leading coefficient from the x² and x terms before completing the square. Skipping this step is the most common source of errors in vertex form when a ≠ 1.
Practice
20 questions across four difficulty tiers. Root-finding questions want
your values comma-separated (e.g. 2, 5). Discriminant
questions want a short phrase, like "no real roots" or "one repeated real
root." Up to three tries per question before the solution is shown.