Topic 2 Summary On Quadratic Functions

Write a quadratic equation in vertex form for the parabola that has a vertex at (-2, 3) and goes through the point (2, 15)

Completing the square requires us to find a value for c that makes a perfect square trinomial in the form

 $f\left(x\right)=a\left(x^2+bx+\left(\frac{b}{2}\right)^2\right)+c-a\left(\frac{b}{2}\right)^2$  

 $f\left(x\right)\ =\ 2x^2+8x+6$    Given
 $f\left(x\right)=2\left(x^2+4x\right)+6$  get x² and x by themselves
 $f\left(x\right)=2\left(x^2+4x+\left(\frac{4}{2}\right)^2\right)+6-8$  balance equation
 $f\left(x\right)=2\left(x+2\right)^2-2$  rewrite as perfect square binomial

When a parabola does not cross the x-axis, we will get a complex number when we complete the square.

 $\ x^2+6x\ +12=0$  
 $x^2+6x=-12$  
 $x^2+6x+\left(\frac{6}{2}\right)^2=-12+9$  
 $\left(x+3\right)^2=-3$  
 $x+3=\pm\sqrt{-3}$  
 $x=-3\pm i\sqrt{3}$  

 $9x^2+36$  
 $\left(3x-6i\right)\left(3x+6i\right)$  
 $9x^2+18i-18i-\left(6i\right)^2$  
 $9x^2-\left(-36\right)$  
 $9x^2+36$  

How do you find the solution to a system of two equations, one being linear and the other a quadratic?

Find the solution to the following equations