Section 10-4
Problem Solving Using the Discriminant
Suppose you want to find the solutions to the equation:
ax2 + bx + c = 0 . . . (1)
This equation can have two, one or zero solutions, depending on the values of a, b, and c. The problem corresponds to finding the x-intercepts of the graph of
y = ax2 + bx + c . . . (2)
If the graph of (2), which is a parabola, crosses the x-axis twice, there are two real solutions to equation (1); if the vertex is on the x-axis, there is one real solution; if the parabola does not cross the x-axis at all, there are no real solutions.
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Parabola with two x-intercepts |
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Parabola with one x-intercept |
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Parabola with no x-intercept |
In the quadratic formula, the expression under the square root sign, b2 – 4ac, is called the discriminant.
If the discriminant is negative, then there are no real solutions. (You need complex numbers to deal with this case properly. These are usually taught in Algebra 2.)
If the discriminant is zero, there is only one solution. If the discriminant is positive, then the ± symbol means you get two answers.
Example 1:
Solve the quadratic equation.
x2– x – 12 = 0
Here a = 1, b = –1, and c = 12. Substituting, we get:
Simplify.
The discriminant is positive, so we have two solutions:
x = 4 and x = –3
In this example, the discriminant was 49, a perfect square, so we ended up with rational answers. Often, when using the quadratic formula, you end up with answers which still contain radicals.
Example 2:
Solve the quadratic equation.
3x2 + 2x + 1 = 0
Here a = 3, b = 2, and c = 1. Substituting, we get:
Simplify.
The discriminant is negative, so this equation has no real solutions.
