Skill: quadratic-equations

Quadratic Equations

A quadratic equation has four doors: factoring, square roots, completing the square, and the quadratic formula. Learn each key — and let the discriminant tell you how many real solutions wait behind the door.

1 Understand

The core idea in plain language.

What it is. A quadratic equation is ax² + bx + c = 0 with a ≠ 0. Factoring splits it and uses zero product. Square roots undo x² = k as x = ±√k. Completing the square rewrites x² + bx as (x + b/2)² − (b/2)². The quadratic formula x = (−b ± √(b² − 4ac)) / 2a always works. The discriminant D = b² − 4ac predicts: D > 0 two solutions, D = 0 one, D < 0 none (real).

Why it matters. Quadratics model every thrown ball, every profit curve, every area with a square in it. Choosing the fastest method — not just any method — is the real skill.

Where it is used. Projectile motion · optimization (max profit, min cost) · geometry with areas.

2 See It

Diagrams that make the idea visual.

Four methods, one equation

x² − 5x + 6 = 0 factors as (x − 2)(x − 3) = 0, so x = 2 or 3. The same answers come from the quadratic formula — factoring is just the shortcut when the numbers are nice.

x² − 5x + 6 = 0 solved four ways — every method lands on x = 2, 3.

The discriminant predicts

D = b² − 4ac is the part under the square root. D > 0: the parabola crosses twice. D = 0: it just touches. D < 0: it never touches — no real solutions.

D > 0: two x-intercepts. D = 0: one (touches). D < 0: none.

3 Worked Examples

Follow each step. The pattern is always the same.

Example 1 Factor: x² + 7x + 12 = 0

Factor: two numbers with product 12, sum 7 → (3, 4): (x + 3)(x + 4) = 0.
Zero product: x = −3 or x = −4.

x = −3 or x = −4

Check: 9 − 21 + 12 = 0 (checks); 16 − 28 + 12 = 0 (checks).

Example 2 Square roots: x² − 9 = 0

x² = 9 — take square roots of both sides.
Remember ±: x = 3 or x = −3.

x = ±3

Check: (±3)² − 9 = 0 (checks).

Example 3 Formula: 2x² + 5x + 2 = 0

D = 25 − 16 = 9; x = (−5 ± 3)/4.
x = −8/4 = −2 or x = −2/4 = −1/2.

x = −2 or x = −1/2

Check: 2(4) − 10 + 2 = 0 (checks); 2(1/4) − 5/2 + 2 = 0 (checks).

Example 4 No real solutions: x² + 2x + 5 = 0

D = 4 − 20 = −16 < 0.
Negative discriminant → no real solutions.

No real solutions

Check: The parabola y = x² + 2x + 5 never touches the x-axis.

Example 5 Complete the square: x² + 6x = 7

Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16.
(x + 3)² = 16 → x + 3 = ±4 → x = 1 or x = −7.

x = 1 or x = −7

Check: 1 + 6 = 7 (checks); 49 − 42 = 7 (checks).

4 Common Mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

Mistake 1: forgetting ±

Wrong

x² = 9 → x = 3.

Right

x = ±3 — every positive square has two roots.

Rule: square-rooting an equation always gives ±.

Mistake 2: formula sign slips

Wrong

For x² − 5x + 6 = 0, writing x = (−5 ± 1)/2.

Right

x = (5 ± 1)/2 — the formula uses −b, and b = −5 so −b = +5.

Rule: −b flips the sign of b. Double it when b is negative.

Mistake 3: solving before = 0

Wrong

Factoring x² + 7x = −12 as x(x + 7) = −12 and “solving” x = −12 or x = −19.

Right

Move everything first: x² + 7x + 12 = 0, then factor: x = −3, −4.

Rule: zero product needs = 0. Rearrange before factoring.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Solve x² − 5x + 6 = 0 by factoring.

Answer

x = 2 or x = 3.

2. Solve x² − 4 = 0.

Answer

x = ±2.

3. How many real solutions does x² + x + 1 = 0 have?

Answer

None — D = 1 − 4 = −3 < 0.

Key Points to Remember

Standard form first: ax² + bx + c = 0 before any method.
Factor when the numbers are nice; zero product gives the roots.
x² = k → x = ±√k — never drop the ±.
Completing the square: add (b/2)² to both sides, then square-root.
Quadratic formula x = (−b ± √(b²−4ac)) / 2a always works — mind the −b sign.
D = b² − 4ac: positive → two real, zero → one, negative → none.
Check each root in the ORIGINAL equation.

Practice problems Try the tool See it on FiboMap