Quadratic Formula

 The Quadratic Formula is a useful formula for solving for x in a quadratic equation such as x2 + x + 1. This formula finds the x-intercepts of the equation. The Quadratic Formula ONLY works for quadratic equations.

The Quadratic Formula states

if ax2 + bx + c = 0.

Example: Solve x2 + 4x - 32 = 0

First, put the coefficients into the quadratic equation, a=1 b=4 c=-32                                           

So, the equation will look like this:

    x= -4 ± √(42) - (4∗1∗-32)

                    2∗1

    x= -4 ± √16- -128

                  2

    x= -4 ± √12

              2

    x= -.2679 and x= -3.7320

The answer means that the equation x2 + 4x - 32 crosses at -.2679 and -3.7320.

Sometimes, there may be no solution to the equation which means that the graph doesn't touch the x-axis at all and the solutions are imaginary. Other times, you get two solutions but they are the same which means that the graph touches in one spot on the x-axis.

Not always will the equation be all set up and ready to go in the ax2 + bx + c =0 so you will have to solve in order to get to it. For example (x+3)2 = 81 doesn't really look like it is a quadratic, but with some work it is. First use the FOIL method:

(x+3)(x+3) =81 (we are doing this because that squared CAN NOT just be distributed)

(x * x) + (x * 3) + (x * 3) + (3 * 3) = 81

x2 + 3x + 3x + 9 = 81

x2 + 6x + 9 = 81

x2 + 6x + (9 - 81) = 0

x2 + 6x - 72 = 0

WOO A QUADRATIC!!!!!!!!!

Now plug this into the quadratic formula:

-6 ± √(62) - 4(1)(-72)

----------------------------  =

                2(1)

-6± √36 + 288

--------------------  =

            2

 -6 ± 18=    -24 = -12    AND         12 = 6

       2          2       1                      2    1

Your x intercepts are -12 and 6.

You can also use the quadratic equation to find "critical points" in inequalities (like greater than, less than, greater than or equal to, etc.)

I'll just use the nice equation I had above but throw in < instead of a = so I won't bore you with another half a page of math...

So here your critical points are -12 and 6. Now you just have to do a simple test and you win! So here it is:

You pick a point in between each of these intervals and plug it into the original equation. If it makes sense then it's a solution, if not it's not a solution. So...

x2 + 6x + 9 < 81 so...

x2 + 6x - 72 < 0

You are looking for negatives here / numbers less than 0

(-∞, -12) , plug in -15... (-15)2 + 6(-15) - 72 = 63, no 63 is positive

(-12, 6) , plug in 0... (0)2 + 6(0) - 72 = 9, yes -72 is negative

(6, ∞) , plug in 10... (10)2 + 6(10) - 72 = 88, no 88 is positive

(-12,6) is your answer (Note: may not see this on the ACT/ SAT but its good to know when you get to college so don't get mad if it doesn't make sense)

Quadratic Formula Song!