The Quadratic Formula and the Discriminant

The quadratic formula is yet another method that can be used to find the solutions to any quadratic equation.

The Quadratic Formula
The solutions of any quadratic equation of the form
ax2 + bx + c = 0 are given by the following formula.
, where .
example #1: / Solve x + 2x – 24 = 0
by using the quadratic formula.
Next, identify a, b, and c values. / a = 1, b = 2, c = –24
Substitute these values into the formula. /
The solutions are 4 and –6.

It is also possible to solve quadratic equations that contain complex roots using the quadratic formula.

example #2: / Solve
by using the quadratic formula.
First, set the equation equal to zero. /
Next, identify a, b, and c values. / a = 1, b = –6, c = 13
Substitute these values into the formula. /
The solutions are 3 + 2i and 3 – 2i.

The Discriminant

The quadratic formula can be used when the coefficients are any complex numbers. Now we want to focus our attention to equations with only real number coefficients. Notice the expression under the radical b2 – 4ac in the quadratic formula is called the discriminant. From this number we can determine the nature of the solutions of a quadratic equation.

Three possible cases:
1. / b2 – 4ac = 0 / Exactly one real number solution exists.
2. / b2 – 4ac > 0 / Two real number solutions exist.
3. / b2 – 4ac < 0 / Two complex solutions exist.

Nature of the Root(s) and Related Graph

By evaluating the discriminant you can also determine the behavior of the graph and the type of root to expect. The table below summarized all the possibilities.

Value of discriminant / Discriminant a perfect square? / Nature of roots / Nature of graph
1. / b2 – 4ac = 0 / --- / 1 real / intersects x-axis once
2. / b2 – 4ac > 0 / yes / 2 real, rational / intersects x-axis twice
3. / b2 – 4ac > 0 / no / 2 real, irrational / intersects x-axis twice
4. / b2 – 4ac < 0 / --- / 2 imaginary / does not intersect x-axis
example #3: / Find the value of the discriminant for each quadratic equation. Then describe the nature of the roots.
/ The value of the discriminant is 22, which is not a perfect square. There are two real, irrational roots. The graph crosses the x-axis twice.
/ The value of the discriminant is 0,so there is one real root. The graph crosses the x-axis once.
/ The value of the discriminant is 36, which is a perfect square. There are two real, rational roots. The graph crosses the x-axis twice.
/ The value of the discriminant is a negative number, so there will be two imaginary roots. The graph will not cross the x-axis.

NOTE: When the value of the discriminant is a perfect square, then the quadratic polynomial can be factored.

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