The values of x and y that satisfy a quadratic equation are called the roots of the equation. The roots of are the x-intercepts of the graph .
In the quadratic formula . the expression is called the discriminant. It gives us information about the roots of the quadratic equation .
Notice that when there are 2 real roots the discriminant .
When there are 2 equal roots (or just 1 real root), .
When there are no real roots, .
We often use .
1. Show that the equation has no real roots.
2. Find the values of k for which the quadratic equation has real roots.
3. Show that for all x.
As a general rule: if two quadratic expressions are equivalent to each other then the corresponding coefficients must be equal.
If for all real x then
1. Find values for a, b and c if .
2. Find the equation of the parabola that passes through the points (-1, -3), (0, 3) and (2, 21).
Sum and Product of Roots
The general quadratic equation can be written in the for
where and are the roots of the equation.
If and are the roots of the quadratic equation ,
Sum of roots:
Product of roots:
1. Find the quadratic equation that has roots 6 and -1.
2. Find the quadratic equation that has roots and .
3. Find the value of k if one root of is -2.
4. Evaluate p if one root of is double the other root.