THE QUADRATIC FORMULAIt works for everything!

…as long as it’s a quadratic equation.

Sick of all these different methods?

Reverse FOIL With a coefficient on x2

Without one Completing the Square

Answers with square roots left in them Answers with imaginary numbers

What are we going to do about it?

Let’s look at the most general form of quadratic formula and solve that for x.

That way we can just plug in every time for “x=“ (since that’s what we’re looking for).

What is the most general form of the quadratic equation?

ax2 + bx + c = 0

Deriving the quadratic formula ax2 + bx + c = 0 Well, normally to solve for x we would

factor…but if we don’t know what the numbers are, we can’t factor.

What do you do if you can’t factor? Complete the square!

Deriving quadform (continued)

ax2 + bx + c = 0 First step in completing the square –

move the constant over. ax2 + bx = -c Now before we do the next step, let’s

divide everything by a – so we don’t have to mess with the really complicated formula for what we add to both sides.

Deriving quadform (continued)

Okay, the thing we add to both sides is what?

Half the x-term, squared.

The whole point of this is to factor it to (x + #)2

Deriving quadform (still)

Okay, now it gets interesting. Before we take the square roots of both

sides, I’m going to multiply everything by 4a2. Do I have to?

No, but it’ll simplify in fewer steps if I do. So:

Deriving quadform (continued)

Multiply through where possible.

Now take the square root of both sides.

Multiply through where possible.

Deriving quadform (the end)

Now we rearrange to solve for x Subtract b from both sides

Divide by 2a

And we’re done!

So how do we use this?

Just plug in the coefficients – a, b, and c – into the equation.

EX: x2 + 8x = 11 x2 + 8x – 11= 0 a = 1 b = 8 c = -11 “Plug and Chug,” as they say.

Example Solution

x = (-8 +\- √(82 – 4(1)(-11)))/(2(1)) x = (-8 +\- √(64 + 44))/2 x = (-8 +\- √(108))/2 Ok, have to use square root knowledge

here 108 = 12*9 = 4*3*9 = 4*9*3, so √(108) = √(4*9*3) = 2*3* √(3) = 6 √(3) x = (-8 +\- 6 √(3))/2 x = -4 + 3√(3), -4 – 3√(3)