Completing the square always works when solving a quadratic equation, but it's almost always a pain in the butt to actually do. We want a method that is faster and easier to use while being just as reliable. What we want is the quadratic formula.

You'll want to memorize this handy little formula. Luckily, it is pretty distinctive. It is possible to derive the formula from scratch, though it requires completing the square (natch). You can see the derivation here, if you like.

### Sample Problem

Solve the quadratic equation 2*x*^{2} – 5*x* – 3 = 0 using the quadratic formula.

We recognize this problem. We did this in the last section—there were all kinds of fractions, and it was nasty to work through. Let's see how it goes with the quadratic formula. To start, we note that *a* = 2, *b* = -5, and *c* = -3. We then set up the equation:

### Sample Problem

Solve the equation 2*x*^{2} + 2*x* + 5 = 0 using the quadratic formula.

Using the quadratic formula is pretty mindless once you know the formula. Just plug and chug and plug and chug and choo choo. Sorry, we got carried away.

That's right, rolling along like a train. Next we check our answers.

Excellent. Now we pull into the station.

## Practice:

Use the quadratic formula to find the roots of *y* = 2*x*^{2} + 4*x* + 1
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How about we check if you know your abc's? *a *= 2* * b = 4
*c* = 1
if (answer = correct) print("Good job!") else { print("Practice some more and you'll surely get the hang of it.") } Next, we plug our numbers into the quadratic formula. Then, simplify. | |

Use the quadratic formula to find the roots of
*
y* = *x*^{2} – 16
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This is a quadratic function, so we can still use the quadratic formula. *a* = 1
*b* = 0
*c* = -16
We just have *b* = 0. Everything works out the same otherwise. And done. | |

Use the quadratic formula to find the roots of
*
y* = *x*^{2} + 4*x* + 4
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*a* = 1
*b* = 4
*c* = 4
Third verse, same as the first. There is only one real root for this equation. | |

Use the quadratic formula to find the roots of . | |

*b* = 3
*c* = 6
In this case we have two complex roots. | |

Use the quadratic formula to find the roots of
*
y* = -*x*^{2} + 5*x* – 2

Hint

Answer

Use the quadratic formula to find the roots of
*
y* = *x*^{2} – 3*x* – 8

Hint

Answer

Use the quadratic formula to find the roots of
*
y* = -4*x*^{2} + *x* – 3

Hint

Answer

Use the quadratic formula to find the roots of
*
y* = *x*^{2} + 3*x
*

Hint

Use the quadratic formula to find the roots of

Hint

Answer