Quadratic Formula

Quadratic Formula

Finding the Roots when Factoring isn't Possible

Where do we get the quadratic formula?

 $ax^2+bx\ +c=0$ 
 $4a\left(ax^2+bx+c\right)=0$ 
 $4a^2x^2+4abx+4ac=0$ 
 $4a^2x^2+4abx\ =-4ac$ 
 $4a^2x^2+4abx+b^2=b^2-4ac$ 
 $\left(2ax+b\right)^2=b^2-4ac$ 
 $2ax+b=\pm\sqrt{b^2-4ac}$ 
 $2ax=-b\pm\sqrt{b^2-4ac}$ 
 $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$  
 

How do we use it?

• First, the equation must be in standard form,

   $ax^2+bx+c=0$  
  

• Next, identify a, b, and c. Substitute them into the formula.
  

• Finally, simplify!
  

Example:  $x^2+2x\ -63=0$  

 $x=\frac{-2\pm\sqrt{\left(2\right)^2-4\left(1\right)\left(-63\right)}}{2\left(1\right)}$ 

 $x=\frac{-2\pm\sqrt{4+252}}{2}$ 

 $x=\frac{\left(-2\pm16\right)}{2}$ 

 $x=7\ or\ -9$  

Practice

 $1.\ \ \ 6x^2-11x-10=0$ 

 $2.\ \ \ -x^2+2x=1$ 

 $3.\ \ \ -2x^2+3x=4x-15$ 

 $4.\ \ \ 8x^2=10x$