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

^{2}+ bx + c = 0, where a, b, c ∈ ℝ and a≠0.

We denote the discriminant by the capital *D*. The discriminant is a real number that equals

^{2}− 4 · a · c.

Take as an example the equation *2x ^{2} − 9x + 7 = 0*. First, we identify the coefficients:

*a = 2*,*b = − 9*and*c = 7*.

Then we can compute the discriminant:

^{2}− 4 · 2 · 7 = 81 − 56 = 25.

Ok, we have the discriminant. What’s it good for? We can infer from it the number of real roots of the equation:

**if**then the equation has two distinguish real roots.*D > 0***if**then the equation has a single real root.*D = 0***id**then the equation has no real root.*D < 0*

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: *x _{1} = 7/2* and

*x*. See the graph of the quadratic function

_{2}= 1*f(x) = 2x*:

^{2}− 9x + 7We can see that the graph intersects the horizontal axis in two points: *(1, 0)* and *(7/2, 0)*.