Solving Quadratic Equations
After working through this section you will:

Quadratic Equations
A quadratic equation is an equation of the form ax^{2}+bx+c=0.
For example, 2x^{2}7x15=0.
So, what does it mean to solve a quadratic equation?
Solving a quadratic equation means finding those values of the variable that make the equation true.
For example, solving 6x^{2}+10x+3=7 means finding the values of x that make this equation true.
If x=2, 6(2)^{2}+10(2)+3=7
If x=1/3,6(1/3)^{2}+10(1/3)+3=7
Only these two values of x: 2 and 1/3 make this equation true, so these are the solutions to the equation 6x^{2}+10x+3=7.
How do I find these values of x that make the equation true?
Before we discuss that, there is an important theorem we need to revisit:
If axb=0 then
 a=0
 b=o
 or both a and b are zero.
Now we have reviewed the Zero Factor Principle, we can continue our exploration into solving quadratic equations. We see now that we can solve any factorequation of the form ab=0.
We will look at a specific example of a factorable quadratic equation 6x^{2}+x15=0 where a = 6, b = 1 and c = 15. Note:the sign on the number must be included.
ax^{2}+bx+c=0  6x^{2}+x15=0  
Step 1: Multiply a by c.  a x c = ac  6 x 15 = 90 
Step 2: Find the factors of ac that add to give b.  90 = 9 x 10; 9 + 10 = 1  
Step 3: Rewrite the equation, represent b as the
sum of the two factors you just found. 
6x^{2}9x+10x15=0  
Step 4: Factor the equation by grouping.  3x(2x3)+5(2x3)=0  
This equation now has two terms that
have a common factor of (2x3), factor again. 
(2x3)(3x+5)=0 
Now you have a factorequation of the form a x b = 0: (2x3)(3x+5)=0
Here we will use the Zero Factor Principle, so
 2x3 = 0, and so solving for x, x = 3/2
 3x+5 = 0, and so solving for x, x = 5/3
Solve the following quadratic equations, using the method shown above:
 x^{2}2x3=0
 6x^{2}x2=0
 2x^{2}x6=0
Solutions
 x^{2}2x3=0
Step 1: 1 x 3 = 3
Step 2: Factors of 3 that add to 2: 1 and 3
Step 3: x^{2}+x3x3=0
Step 4: x(x+1)3(x+1)=0
(x+1)(x3)=0
x+1=0 and x3=0
Therefore, x = 1 and 3