The quadratic formula and choosing a method

For \(ax^2+bx+c=0\), \(a\ne0\), the quadratic formula is \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\). Include the signs of the coefficients, and put the entire numerator over \(2a\).

Example

Solve \(2x^2-3x-1=0\). Here \(a=2\), \(b=-3\), \(c=-1\).

\(\begin{aligned}x&=\dfrac{-(-3)\pm\sqrt{(-3)^2-4(2)(-1)}}{2(2)}\\&=\dfrac{3\pm\sqrt{17}}4.\end{aligned}\)

Thus \(x\approx1.78\) or \(x\approx-0.281\), to 3 significant figures. Keep the unrounded values until the final step.

Choose factorisation when factors are easy to see, completing the square for a monic expression or when asked, and the formula when factorisation is awkward. If the number under the square root is negative, that equation has no real solution.