Quadratic formula is one of 4 quadratics formulas in the algebra section of this library, and it is used at high school · ap level.
Why quadratic formula works
It is completing the square carried out once with letters instead of numbers. Dividing through by a and squaring off the x terms turns the equation into a perfect square set equal to (b squared minus 4ac) over 4a squared. Taking the square root of both sides produces the plus-or-minus and the 2a underneath.
What each symbol means
$a,b,c$ are coefficients of $ax^2+bx+c=0$.
Quadratic formula: when it holds
$a\ne0$; keep the entire numerator over $2a$.
When it stops applying
If a is zero there is no x squared term and 2a is zero, so the division is impossible. The equation bx + c = 0 is linear and is solved in one step by x = -c/b. Always confirm a nonzero leading coefficient before reaching for this formula.
Quadratic formula: a worked example
For $2x^2+3x+6=0$, $x=\frac{-3\pm i\sqrt{39}}{4}$.
The mistake to avoid
What people do: Writing x = -b plus or minus the square root, all divided by 2a only under the radical.
Why it goes wrong: The -b belongs over 2a as well. Dividing only part of the numerator gives answers that do not satisfy the original equation when you substitute them back.
Do this instead: Draw the full-width fraction bar before you write anything above it, so both the -b and the radical land on top of the same denominator.
Quadratic formula: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,b,c$ are coefficients of $ax^2+bx+c=0$.
- Check the conditions before substituting. $a\ne0$; keep the entire numerator over $2a$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Quadratics
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where quadratic formula comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- The Quadratic Formula Explained — the lesson behind this formula: use the formula and interpret its discriminant.
- Quadratic Formula Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about quadratic formula
Does it still work when the equation has no constant term?
Yes, c is simply zero. For x squared minus 5x = 0 the formula gives (5 plus or minus 5) over 2, which is 5 and 0, the same answers factoring produces.
Do I have to move everything to one side first?
Yes. The letters a, b and c are only meaningful once the equation reads ax squared plus bx plus c equals zero, so collect all terms on the left before identifying them.
Why does it sometimes give the same answer twice?
Because the quantity under the square root came out to zero, so adding and subtracting it changes nothing. Both roots land on -b over 2a and the parabola touches the axis at a single point.
Is factoring ever faster than this?
Often, when the roots are small whole numbers. The formula is the reliable fallback: it works on every quadratic, including the ones with irrational or complex roots that will never factor nicely.