Why does the discriminant b^2-4ac determine the number of roots of the quadratic equation ax^2+bx+c=0?
The rule for the discriminant: if b^2-4ac>0 then the quadratic has two roots if b^2-4ac=0 then the quadratic has one root if b^2-4ac<0 then the quadratic has no rootsRecall that the formula for solving the quadratic equation ax^2+bx+c=0 is x=(-b+(b^2-4ac)^0.5)/2a. Notice that the square-root of the discriminant is contained in this formula. If the discriminant is positive then it has a positive and negative square-root, giving two possible roots of the equation. If the discriminant is zero then this square-root term disappears giving only one root to the equation. Lastly, if the discriminant is negative then this square-root does not exist so the formula gives no answer.
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