Quadratic Equations
“A quadratic's solutions are where the parabola meets the x-axis.”
The formula
ax² + bx + c = 0 → x = (−b ± √(b² − 4ac)) / (2a)How to read it: the two roots of a quadratic come from plugging a, b, c into the quadratic formula
- a, b, c
- — the coefficients of the x², x, and constant terms (a ≠ 0)
- b² − 4ac
- — the discriminant — it decides the number of roots
- ±
- — the sign giving both roots at once
The hook
Some quadratics won't factor — and when they won't, the quadratic formula is the master key that always works.
In plain words
An equation with an x² in it, ax²+bx+c=0. It has at most two solutions, and plugging a, b, c into the quadratic formula always produces them.
The intuition
The graph of y=ax²+bx+c is a parabola. The x-coordinates where it crosses the x-axis are exactly the equation's roots — cross twice for two roots, graze for one, miss for no real roots.
How it's built
If it factors, use 'one of the two brackets must be 0' to solve fast. If not, use the quadratic formula. The b²−4ac under the root (the discriminant) tells you the number of roots — positive gives two, zero gives one, negative gives no real roots.
Example
x²−2x−3=0 → (x−3)(x+1)=0 → x=3 or x=−1, because for a product to be 0, one of the brackets must be 0.
Common mistake
'If one factor is 0 the product is 0' only works when the other side is 0 — you can't use it on (x−1)(x−2)=6 (rearrange to 0 first). And if the discriminant is negative there's no (real) solution.
Where it's used
The height of a thrown object, the side of a plot with a fixed area, the point of maximum profit — the moment a square appears, a quadratic shows up.
Where it came from
The roots of the formula reach back to ancient Babylonian area problems; the full algebraic formula was polished across many civilizations.
Prerequisites
Quick check
What are the two roots of x² − 5x + 6 = 0?
- 2 and 3✓
- −2 and −3
- 1 and 6
- 2 and −3
Practice
Find the positive root of x² − 4 = 0.
Answer: 2
- x² = 4
- x = 2 or x = −2
- the positive root is 2
Takeaway: x²=k gives ±√k — keep both signs.
Find the larger root of x² − 7x + 10 = 0.
Answer: 5
- Two numbers multiplying to +10 and adding to −7: −2 and −5 → (x − 2)(x − 5) = 0
- x = 2 or x = 5
- the larger root is 5
Takeaway: If it factors, you solve fast without the formula.
Find the root (a repeated root) of x² − 6x + 9 = 0.
Answer: 3
- (x − 3)² = 0
- x = 3 (the two roots coincide)
Takeaway: A perfect square gives one repeated root (discriminant 0).
Solve x² + 2x − 1 = 0 with the quadratic formula.
Answer: undefined
- a=1, b=2, c=−1 → b²−4ac = 4 + 4 = 8
- x = (−2 ± √8)/2 = (−2 ± 2√2)/2
- x = −1 ± √2
Takeaway: When it won't factor, the formula always delivers.