How to use the discriminant and sum and product of roots

The Quadratic Formula helps if you have difficulties factoring, or if the quadratic equation can not be factored, nevertheless; what if I need to find the solutions of a quadratic equation, but this doesn't have real solutions? Is there a way of avoiding to go all the way till the end of the process for the Quadratic Formula? Is there a shortcut? ... Once I have found the solutions of a quadratic equation, is there a form in which I may verify it?

In this detailed lesson, you will learn those shortcuts, and how you may verify the solution of a quadratic equation. You will be sad when the lesson comes to an end!

Lesson's Content


Lesson In PDF Format (no animations)

Lesson's Glossary

Discriminant: In a quadratic equation is the expression inside the radical: b2 - 4ac.

Function: A relation of the type that has exactly one value in the domain (independent variable) matching a value in the range (dependent variable).

Function notation: A function written with the symbol f(x) instead of y. It is read as f of x.

Quadratic equation: An equation of the form

ax2 + bx + c = 0 where a, b, and c are real numbers and a is different from zero.

Quadratic formula: If ax2 + bx + c = and a is different from zero then the quadratic formula is given in terms of a, b, and c.

Quadratic function: Any function in the form of

f(x) = ax2 + bx + c where a is different from zero. The graph is a parabola and the largest exponent is 2.

Solution or root: The value that makes an equation a true statement, a root refers particularly to the value of x for which y = 0, this value is also the x-intercept of the graph.

Vertex: The lowest point for a parabola that opens up (minimum); the highest point for a parabola that opens down (maximum).

Vertex form of a quadratic function: The vertex form of a quadratic function is: f(x) = a(x-h)2 + k. The vertex coordinates are (h,k).

Zeros of a function: The solutions for the equation of the function when this equal to 0. The roots, also known as the x-intercepts.

Interactive Algebraic Applets

A quadratic function may have two real zeros, one real zero, or two imaginary

zeros. This is determined by the discriminant. If the discriminant is positive, then

you have two real solutions; if the discriminant is zero, then you have one real

solution; and if the discriminant is negative, then you don't have real solutions.

This interactive algebraic applet helps you to visualize these relationships.

Drag any of the sliders to generate a different graph, with different solutions.

Didn't you find what you were looking for? Do your search here!

Discriminant of the Quadratic Formula. Sum and Product of Roots.

In this detailed lesson, you will learn those shortcuts, and how you may verify the solution of a quadratic equation. You will be sad when the lesson comes to an end!