Same Math › Algebra 1 · grades 8–10 › The quadratic formula
How do kids learn the quadratic formula?
The quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a, solves any equation ax² + bx + c = 0. Kids write down a, b and c, find the discriminant b² − 4ac first, and then plug in. Many classes sing it to the tune of “Pop Goes the Weasel” so it sticks.
- Label a, b and c first, signs included.
- The discriminant b² − 4ac says how many real solutions there are.
- ± means two answers: one with +, one with −.
Why schools teach it
It always works
Students solve quadratics by inspection, square roots, completing the square, factoring and the formula, and pick the method that fits (HSA.REI.B.4b). The formula works even when factoring doesn't.
Classes teach the discriminant first because it catches most mistakes early: if b² − 4ac is negative, there are no real answers, so there's nothing to compute.
At the kitchen table
What to say, and what to skip
Say this
What are a, b and c? Watch the signs.
What's the discriminant? Is it positive, zero or negative?
What's −b here?
If b is −6, then −b is 6.Can the square root be simplified?
Skip this
- Squaring a negative b without parentheses.If b = −3, b² is (−3)² = 9, not −9.
- Dividing only part of the top by 2a.The whole top, −b ± √…, goes over 2a.
- Forgetting to set the equation equal to zero.x² + 3x = 4 needs to become x² + 3x − 4 = 0 first.
See it in action
Solving 2x² − 3x − 2 = 0
a = 2, b = −3, c = −2
Discriminant
(−3)² − 4(2)(−2) = 9 + 16 = 25Plug in
x = (3 ± 5) ÷ 4Two answers
x = 2 or x = −1/2
A perfect-square discriminant means the equation could also be factored.
Free practice sheet
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Parents also ask
What's the song?
To “Pop Goes the Weasel”: x equals negative b, plus or minus the square root, of b squared minus four a c, all over two a. Kids remember it for years.
What does a negative discriminant mean?
No real solutions: the parabola never crosses the x-axis. In Algebra 2 these become complex numbers.
Do answers have to be simplified?
Usually yes: simplify the square root and reduce the fraction. Some teachers also want decimals rounded.
When should kids use the formula instead of factoring?
When factoring isn't quick. If the discriminant is a perfect square, factoring would have worked too.