We already know how to solve quadratic equation that has no linear term or no constant. But how to solve a quadratic equation that has all its terms? One way – the ultimate one – is to compute the discriminant.
Recall that the quadratic equation is an equation in the form
We denote the discriminant by the capital D. The discriminant is a real number that equals
Take as an example the equation 2x2 − 9x + 7 = 0. First, we identify the coefficients:
- a = 2, b = − 9 and c = 7.
Then we can compute the discriminant:
Ok, we have the discriminant. What’s it good for? We can infer from it the number of real roots of the equation:
- if D > 0 then the equation has two distinguish real roots.
- if D = 0 then the equation has a single real root.
- id D < 0 then the equation has no real root.
Now we can use the discriminant to compute the roots. We just use another magic formula:
As you can see, we used the ± symbol to denote that the formula produces two outcomes: one with plus sign and one with minus sign. We can rewrite the formula this way (see the sign before the discriminant):
Now we can substitute all the coefficients and we can compute all the roots of our equation:
The first root equals 7/2. What about the second?
The second root is a lot of nicer, it’s just one. In the end we’ve got our two roots: x1 = 7/2 and x2 = 1. See the graph of the quadratic function f(x) = 2x2 − 9x + 7:
We can see that the graph intersects the horizontal axis in two points: (1, 0) and (7/2, 0).