How to know how many solutions a quadratic equation has

The discriminant is the term underneath the square root in the quadratic formula and tells us the number of solutions to a quadratic equation. If the discriminant is positive, we know that we have 2 solutions. If it is negative, there are no solutions and if the discriminant is equal to zero, we have one solution. The discriminant is calculuated by squaring the "b" term and subtracting 4 times the "a" term times the "c" term.

The discriminant is a really handy tool when you think you're getting a weird answer. Here's why. The discriminant tells you how many solutions there are to quadratic equation or how many x intercepts there are for a parabola. It doesn't tell you what those numbers are like what the x intercept values are, it just tells you how many of them there should be. And it sounds like that's not useful but it actually is especially when you're checking your work.
So here is what it looks like. The discriminant is the formula b squared minus 4ac remembering that a, b and c are the coefficients of your quadratic in standard form. It tells you the number of solutions to a quadratic equation. If the discriminant is greater than zero, there are two solutions. If the discriminant is less than zero, there are no solutions and if the discriminant is equal to zero, there is one solution.
This is something you just kind of have to memorize. It goes hand in hand with the quadratic formula. So if you guys have learned that, this will make a lot of sense. If you haven't learnt the quadratic formula yet you'll probably learn it tomorrow in math class. Just know that, what you're looking at is whether or not b squared minus 4ac is greater than zero, less than zero or equal to zero. And it tells me how many answers I should have. It doesn't tell me what the answers are just how many of them I should have in order to get the problem correct.

The Quadratic Formula

Trinomials are not always easy to factor. In fact, some trinomials cannot be factored. Thus, we need a different way to solve quadratic equations. Herein lies the importance of the quadratic formula:

Given a quadratic equation ax2 + bx + c = 0, the solutions are given by the equation

x =

Example 1: Solve for x: x2 + 8x + 15.75 = 0

a = 1, b = 8, and c = 15.75.
x =

=

=

=

=

or

= -

or-

Thus, x = -

x = -

Example 2: Solve for x: 3x2 - 10x - 25 = 0.

a = 3, b = - 10, and c = - 25.
x =

=

=

=

=

=

or

= 5 or-

Thus, x = 5 or x = -

Example 3: Solve for x: -3x2 - 24x - 48 = 0.

a = - 3, b = - 24, and c = - 48.
x =

=

=

=

=

=

= - 4

Thus, x = - 4.

Example 4: Solve for x: 2x2 - 4x + 7.

a = 2, b = - 4, and c = 7.
x =

=

=

=

Since we cannot take the square root of a negative number, there are no solutions. (The graph of this quadratic polynomial will therefore be a parabola that never touches the x-axis.)

The Discriminant

As we have seen, there can be 0, 1, or 2 solutions to a quadratic equation, depending on whether the expression inside the square root sign, (b2 - 4ac), is positive, negative, or zero. This expression has a special name: the discriminant.

How do you know if a quadratic equation has one solution?

How do you tell if a quadratic equation has 0 1 or 2 solutions?