The Discriminant and the Number of Solutions
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The discriminant D = b² − 4ac tells you how many solutions there are before you solve anything. D > 0 means two solutions, D = 0 one, D < 0 none. It also finds parameters: x² + mx + 9 = 0 has one root when m² − 36 = 0, so for m = 6 or m = −6.
What does this generator do?
On the easy level you calculate the discriminant, including negative coefficients. On the medium level you use it to decide whether there are two solutions, one or none. On the hard level you find the values of m that give a repeated root, for example in x² + mx + 16 = 0 or mx² − 4x + 1 = 0.
How to use it
1) Write down a, b and c. 2) Substitute into D = b² − 4ac, with negative numbers in brackets. 3) D > 0 means two solutions, D = 0 one, D < 0 none. 4) With a parameter, write D in terms of m, set it equal to zero and solve. 5) If m multiplies x², check that m = 0 is not an answer — then the equation would not be quadratic.