The discriminant
The quadratic formula carries a small expression under the square root: b squared minus 4ac. Its sign decides how many real solutions exist, so a question about the number of roots can be answered without solving. Read the sign and the work is done.
The part under the root
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. The expression b² - 4ac is called the discriminant, and it decides what the square root does. Every other part of the formula stays the same, so the discriminant is the switch that changes the outcome from two answers to one or to none.
Positive, zero, negative
If b² - 4ac is positive, the square root is a real number and the plus minus symbol gives two distinct real solutions. If it is zero, the two branches collapse into one repeated solution. If it is negative, there is no real square root, so the equation has no real solutions.
How the test uses it
These questions rarely ask you to solve. They ask how many real solutions an equation has, or for the value of a constant that makes the count turn out a certain way. Set b² - 4ac equal to zero to find the boundary, then reason about which side of it the question sits on.
Watch the sign of a
Before reading a, b and c, arrange the equation as ax² + bx + c = 0. A missing constant term means c is zero, which is a real value rather than a blank to skip. Getting the signs right matters more than the arithmetic, because one flipped sign changes the answer.
What is the discriminant?
It is b² - 4ac, the expression under the square root in the quadratic formula. Its sign tells you how many real solutions the equation has.
What does a positive discriminant mean?
There are two distinct real solutions. The square root is a real number and the plus minus symbol produces two results.
What does a zero discriminant mean?
There is one repeated real solution. The two branches of the formula meet at a single value.
What about a negative discriminant?
There are no real solutions. A negative number has no real square root, so the equation does not cross the x axis.
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