Some quadratics don't factorise into neat brackets — try x² + 4x + 1 and no whole numbers multiply to 1 and add to 4. The quadratic formula is a tool that solves any quadratic written as ax² + bx + c = 0, factorising or not.
Why this matters: in the real world the numbers are rarely tidy — the answer to "when does the ball hit the ground?" is often a messy decimal. The formula never fails: feed it a, b and c and it hands back the solutions every time, whether they're whole numbers, surds or don't exist at all.
The quadratic formula solves any ax² + bx + c = 0
x = ( −b ± √(b² − 4ac) ) / 2a
The ± ("plus or minus") is why a quadratic usually has two solutions — you work it out once with + and once with −.
Using it — three steps
Write the equation as ax² + bx + c = 0 and read off a, b, c.
Substitute into the formula carefully (mind the signs).
Work out both answers — with + and with −.
x² + 5x + 6 = 0: a=1, b=5, c=6. b²−4ac = 25 − 24 = 1. x = (−5 ± √1)/2 = (−5 ± 1)/2. Plus: (−5 + 1)/2 = −2. Minus: (−5 − 1)/2 = −3. So x = −2 or x = −3.
The ± splits into two branches → the two solutions.
The discriminant (b² − 4ac)
The discriminant b²−4ac decides: cross twice, just touch, or miss.
The part under the root is called the discriminant. Before you finish the calculation, it tells you how many solutions to expect — because you can't square-root a negative number (no real answer), and √0 gives just one value.
b² − 4ac
Number of real solutions
What the graph does
positive
two
crosses the x-axis twice
zero
one (repeated)
just touches the x-axis
negative
none (no real roots)
never reaches the x-axis
Completing the square
Another way to rewrite a quadratic: x² + bx becomes (x + b/2)² − (b/2)². Analogy: you're building the perfect square (x + b/2)², which has a little too much on the end, then subtracting the extra back off. It's used to solve quadratics and to find the lowest (or highest) point of the curve.
x² + 6x + 5: half of 6 is 3, so start with (x + 3)². But (x + 3)² = x² + 6x + 9 — that's 9, and we only want 5, so subtract 9 and add 5: x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4.
(x+3)² builds a perfect square but adds an extra 9 — subtract it back.
Common mistake: −b means "the opposite of b". If b is negative, −b becomes positive — so for x² − 7x + 12 you use −(−7) = +7, not −7. Watch this sign flip carefully; it's the classic slip.
In a nutshell: read off a, b, c; substitute into x = (−b ± √(b²−4ac))/2a; the ± gives two answers; the discriminant tells you how many before you even finish.