The Quadratic Formula
xBZqdZD2quD3xeK2bUWbknAMbidUToWrUPdmKOxjMqhgoNWbHptFrHm06YIPQdum4qqBkGXjeOO9vin0kKEbMUJRxnkLGCwN5tqXrpKSijQ0BBwx/PzG8ewqP/3r+nQcJFrd29orfr+LPgEDzypX298fbOTm3g33KwFhifj3mOdlYQANV1yMjQQ+j74JiQb11HnuiyU0TFC6xw6lDfpHszCBp3p/6TWOvAwwNeruo9OkARRndVoepJWtdu0NKoAFt1pLNnfJdE2NBDssM4TscArIjQQyBM5TIgO5dnwgFGYSgzTQ9A9tZwJQVX3WOe2CfDhkf6vPARZORDU3G6A2fNzsquXKgcHfv51I13Jrb90I1T/6mg/ogh3lgwAxgFtmtKJcSTwCG4NiWNBU2LwC3Kxj1C7m3t1Qh7Qmduwrskq6/JRWn1leUZNlake5uHmJHV3BndNrPEfxZSDvFY822H3XX9miVKMK0WwiaCuzZ7tgxkxK3TZGTE66Msm+YyX1JLzb5czpld4hrTFwdjS91QJv4uak2wxbFOiXydUqxDjy1L2tXgX3AnO2TukqWjpI6jA++lpBjOqollFVKLtTg6BxSgJj1ozxvvM5evTihWXd4Yo5+eAAsIyT9/9XGwNABlY2ARSggq+FKe8fgVizxgCQrAr377+FkSvW8oV3TY9cRoSTPuGMYWpKKgceoAdUaIPRYU86WhaQcbOYbKmivtNf0mI0DiPBZ6J0fi3atxtxdxmEmlKzwAWtvcc9/CCAuLqXp0wBKAE8SNu7TSOJZespEGnk1Cy1SlC9ddTjIh5E8kZ8o2SPsWNLkEQk4UM7t8AtYDmoy+5MoDioFa137fDHcdCuyL5s3jKXTz4maj5GDS+nhzgHyKIqu/cTyyAUn+/wo/yD07/Wlf8yyKr3HGK3TG6fbn+usGJVfARC8XcQg9yJshURtTx8gr5hvYalRzSfiJz4cLRUar6VZdn6/UKBrycKJUBYJ21NxexSesNzPH/Cgm1NAzqDrCzDx9iiKL4o7E7FfE5PzPxcSyDZn0Yb8gd2VvcGTraDo6tU7bAE54k1IpBfnhAy4BRVlNPmbFVeTKjVXbaui30fUuvpDSPzh9lzPbHS3PWa+FzlgYVLkGopjIH/1lZpmEKyu0PB+iF/GP6OIapRZ2rs97BASSG8KXGSoSWDqRI1iMFHsQtUIbIspfHumthRYdsOk0DhYw9K2U/7r+gR328ytFEtLzeiWl4vzQYQBSRyXchtixvTHe2F72ooVRiEuJo4IiWMA171KgAHcRrF0e7mzSqAQ/gO7i/PPYt2ZAq4q652AmE4AzUXV5L0a8h1cuWtYm4QnFV6+d+zwmg/dvxmbbSl3ZP1nrv4LxkTQnct5pGJxIMpxZs79V8DxD38FlCgzOa290qqLFkX2ac7ZUrzu2HC07ykhIGlABq16RQ2sQ/v6gkGNG2aT5H2sdR8e1q6KZfCGSw8jJV4lRbVL55jto9+3iMBtNE96IQKJcXfupAfLY/CIArmo6bH5j56QeFmJZKzZXn4yOdrP3ZRcnKrjlwi7BMJACsJWf8akcJNg1cjCkgdfve9zxs7XydMupZxL8hun7bsDsBK7KW+qvrmZmWKm5liRnuP6TZ1Rr3znu3pBF2jrOfLqslGdY5vrQXTL8COnplFnrg3AScKU5voJqllY+cpeHJnnF/77MoNAf8j9bVVijPB5gdwa/d7+HZ5y/+avGzdzqKcTylTC33bfvZE344tlJZx19w3VDD28LRtaOOOM/WfJCaI+h9B8hZy8zwyhSUYif+Oe7tyHq3qbbGgwjov4KXu7IX6UaV6XyZyQ3N2oBXwLYO6T2jrF+d3VBkHPvbtvPtaZQZkmsgN3CoA5XwFor ax² + bx + c = 0:
The value b² − 4ac (the discriminant) determines the nature of the roots:
- > 0 — two distinct real roots
- = 0 — one repeated real root
- < 0 — two complex conjugate roots