The Quadratic Formula

April 7, 2021 By Xander Polynomials

The quadratic formula is used to solve for the roots of a quadratic polynomial (the highest power is 2)

given $ax^2+bx+c=0$ , where $a\ne0$

Python x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} 1 x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Proof using completing the square

Python ax^2+bx+c=0 1 ax^2 + bx + c = 0

multiplying both sides by $4a$ , we get

Python 4a^2x^2+4abx+4ac=0 1 4a^2x^2 + 4abx + 4ac = 0

Python (2a)^2x^2+(2a)(2b)x+b^2=b^2-4ac 1 (2a)^2x^2 + (2a) (2b) x + b^2 = b^2 - 4ac

Python (2ax+b)^2=b^2-4ac 1 (2ax + b)^2 = b^2 - 4ac

Python 2ax+b=\pm\sqrt{b^2-4ac} 1 2ax+b=\pm\sqrt{b^2-4ac}

Python 2ax=-b\pm\sqrt{b^2-4ac} 1 2ax=-b\pm\sqrt{b^2-4ac}

Python x=\frac{\pm\sqrt{b^2-4ac}}{2a} 1 x=\frac{\pm\sqrt{b^2-4ac}}{2a}