Not all quadratic equations have real roots. There are 3 possibilities for an equation

Whether a quadratic will have real roots can be determined by looking at the square root part of the quadratic formula.

Since you cannot take the square root of a negative number, if 

then there can be no real solutions to the equation.

If the term to be square rooted is positive, then there is no problem. So if

then there will be two real solutions.

We get 2 solutions to the equation because the square root can be positive or negative. However, should the square root be equal to zero, then this would not make any difference. There would therefore only be one real solution. 

The results can be summarised as follows

Examples

For the following quadratic equations, determine how many real roots there are.

Example 1

Define a, b and c and put these into the formula.

Therefore the equation has 2 real distinct roots.

Example 2

Define a, b and c and put these into the formula.

Therefore the equation has no real roots.

Example 3

Define a, b and c and put these into the formula.

Therefore the equation has one real root.

Example 4

Define a, b and c and put these into the formula.

Therefore the equation has no real roots.

Example 5

Define a, b and c and put these into the formula.

Therefore the equation has no real roots.

Without solving a quadratic equation, it is therefore possible to determine some characteristics of its roots.

When a quadratic has 2 real roots you can also determine some extra information about the roots. We will look at this on the next page.

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