The Quadratic Formula

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We now can factor almost any quadratic (the exception is quadratics which result in the square root of a negative number -- that gets into complex numbers!). But it sure would be nice if there was a simple formula we could use instead of going through this whole rigmarole every time.

Well, there is one! It's called the Quadratic Formula. Let's see if we can figure out how it works. First, let's start with a quadratic as general as we can make it. We won't put any numbers in; instead we'll use the variables a, b and c to represent the coefficients:

ax² + bx + c = 0

Even though there are no numbers, we can still do our process, and when we're done, we'll have a really nice formula we can use in the future.

Since a might not be a perfect square, let's multiply both sides of the equation by a:

a²x² + abx + ac = 0

To complete the square, we need to have a constant term of (b/2)² or b²/4. So we add b²/4 - ac to both sides:

a²x² + abx + ac + b²/4 - ac = 0 + b²/4 - ac
a²x² + abx + b²/4 = b²/4 - ac
(ax + b/2)² = (

(b²/4 - ac)

)²
(ax + b/2) = ±

(b²/4 - ac)

ax = -b/2 ±

(b²/4 - ac)

A little bit of algebraic simplifying reduces this to:

This looks a bit intimidating, but it's a very valuable tool. For any quadratic, a is the coefficient of x², b is the coefficient of x, and c is the constant term. Take a look at the example following:

Problem #1
Solve for x if x² + 4x + 2 = 0. Round your answer to the nearest hundredth.

Solution #1
In this problem, a = 1, b = 4, and c = 2.

Plugging these into the Quadratic Formula gives us

x = (-4 ±√(16 - 8))/2

Simplifying these using your calculator results in:

x = -0.59 or x = -3.41

Problem #2
Solve for x if x² - 7x + 5 = 0. Round your answer to the nearest hundredth.

Solution #1
In this problem, a = 1, b = -7, and c = 5.

Plugging these into the Quadratic Formula gives us

x = (7 ±√(49 - 20))/2

Simplifying these using your calculator results in:

x = 6.19 or x = .81

Solve the following, giving your answer rounded to the hundredths place

[ADMINISTRATOR'S NOTE: For additional quadratic problems, click the PRINTABLES link at the top of the page, and then select the QUADRATICS worksheet]