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Algebra · Quadratic equations

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The quadratic formula

Where the formula comes from, how to substitute negative coefficients without sign disasters, and what the value under the square root says about how many solutions exist.

Algebra · Quadratic equations

The quadratic formula

Where the formula comes from, how to substitute negative coefficients without sign disasters, and what the value under the square root says about how many solutions exist.

Why it works

Most quadratics don't factorise nicely — the roots of x2+4x−9=0x^2 + 4x - 9 = 0 are not whole numbers, so no integer factor pair exists to hunt. The formula solves every quadratic ax2+bx+c=0ax^2 + bx + c = 0 in one move:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

It isn't magic: it is completing the square done once, in general, on ax2+bx+c=0ax^2 + bx + c = 0 — the answer to *"what does the square-completing recipe always give?"*, memorised so you never have to redo it.

Substitute with brackets, or the signs will bite. For 3x2−8x+2=03x^2 - 8x + 2 = 0: a=3a = 3, b=−8b = -8, c=2c = 2. Then
  • −b=−(−8)=+8-b = -(-8) = +8 — the formula's minus reverses bb's sign, it doesn't
copy it;
  • b2=(−8)2=+64b^2 = (-8)^2 = +64 — a square is never negative;
  • b2−4ac=64−24=40b^2 - 4ac = 64 - 24 = 40.
So x=8±406x = \frac{8 \pm \sqrt{40}}{6}. When cc is negative the same care pays: −4ac-4ac becomes plus, because subtracting a negative adds.

The whole numerator sits over 2a2a. −b+b2−4ac-b + \sqrt{b^2 - 4ac} is ALL divided by 2a2a — a classic slip divides only the root. Type it into a calculator with brackets around the entire top.

The ±\pm is why quadratics have two solutions: one root from ++, one from −-. And the number under the square root, b2−4acb^2 - 4ac, decides everything before you finish: positive → two solutions; zero → exactly one (the ±\pm adds and subtracts nothing); negative → no solutions at all, because no real number squares to a negative. If an exam asks why an equation has no solutions, compute b2−4acb^2 - 4ac and show it is negative.

Exact or rounded? "Give your answers to 2 decimal places" is a strong hint to use the formula. "Give your answers in surd form" means leave the x\sqrt{\phantom{x}} in, simplified — no calculator decimals.