The discriminant
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How many real solutions does x^2 - 5x + 3 = 0 have?
Common mix-up
In this example, a student wrote b^2 - 4ac = 25 + 12 = 37, so two real solutions..
The student treated -4ac as +4ac and computed 25 + 12 = 37, ignoring the subtraction that is part of the formula.
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How many solutions does x^2 - 5x + 3 = 0 have?
Before solving a quadratic, the discriminant can tell you how many real answers it will have. One number, three cases.
b^2 - 4ac: positive means two real solutions, zero means one, negative means none.
The quadratic formula carries everything you need to count solutions in one expression: b^2 - 4ac. That expression is called the discriminant. Read a, b, and c straight from the standard form ax^2 + bx + c = 0, so for x^2 - 5x + 3 = 0 you have a = 1, b = -5, and c = 3.
For this equation the discriminant is (-5)^2 - 4(1)(3) = 25 - 12 = 13. The most common arithmetic slip is to add 4ac instead of subtracting it, which gives 37 here. The minus sign belongs to the whole term: the formula takes b^2 and then subtracts the product of 4, a, and c.
The sign of the result decides everything. A positive discriminant means the square root in the formula is a real number, and the plus or minus produces two distinct real solutions. A zero discriminant produces exactly one, and a negative discriminant means no real solutions, because no real number squares to a negative.
The discriminant answers the question before you spend any time solving. A positive 13 here tells you the parabola crosses the x-axis twice, and the exact solutions are x = (5 +/- sqrt(13))/2. Substituting either value back gives zero, confirming both crossings.
Keep this idea
Compute b^2 - 4ac before solving. Its sign tells you how many real solutions exist, without solving the equation at all.
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