Quadratic Formula Examples

Solve: 3x² + 2x + 1 = 1.

We need to rewrite the equation so it has zero on one side. Subtract 1 from each side for starters:

3x² + 2x = 0

Now we plug these numbers into the quadratic formula:

Here we go:

So here's our first solution:

And here's the second one:

In this instance, we could also have found the solutions by factoring, since

3x² + 2x = x(3x + 2)

Thankfully, we get the same solutions whether we factor or use the quadratic formula. Once again, we're assured that all is right in the mathematical universe. Boy, we'll sleep well tonight.

Solve: bx² + cx + a, where a, b, and c are real numbers and b is not zero.

Just when you thought you had this whole thing figured out, we go and switch the letters around on you. How terrible are we?

Not actually that terrible, as it turns out; there's a reason we're being so evil. A problem may try and trick you at some point by pulling a similar stunt, and we want you to be well prepared for it. In the quadratic formula, a usually stands for the coefficient on the x² term, as you know, but now we need to write b instead. Usually b is the coefficient on the x term, but now we need to replace every b in the quadratic formula with c. Finally, where the c usually appears in the quadratic formula we need to write a, since that's the constant term now. Bre we cbsiablly sbying thbt alebrly?

We ultimately have

.

Solve: x² + x + 9 = 0.

We use the quadratic formula, of course:

.

Since the discriminant 1 – 4(9) = -35 is less than zero, we can't take the necessary square root, so this equation has no real number solutions. We also do the same thing we do when the outdoor temperature is less than zero: absolutely nothing.