The Quadratic Formula and the Discriminant
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Every quadratic equation ax² + bx + c = 0 can be solved the same way. First work out the discriminant D = b² − 4ac. If it is positive, there are two roots x = (−b ± √D) / 2a. If it is zero, there is one root. If it is negative, there are no real roots.
What does this generator do?
On the easy level the roots are whole numbers, sometimes one repeated root. On the medium level the roots are fractions, for example 2x² − x − 3 = 0, and now and then there is no solution. On the hard level the discriminant is not a square number, and you write the roots as p ± q√r, for example x = 2 ± √3.
How to use it
1) Read off a, b and c with their signs. 2) Work out D = b² − 4ac — put a negative b in brackets. 3) The sign of D tells you how many roots there are. 4) Substitute into x = (−b ± √D) / 2a and simplify. 5) For surd roots, simplify the square root and cancel the fraction.