# Given #x^2 + 4x - 32 = 0# use geometric construction to determine roots? Generalize to #ax^2+bx+c=0# finding a geometric interpretation of a quadratic formula #(-b+- sqrt(b^2-4ac))/(2a)#?

##### 1 Answer

Please see below.

#### Explanation:

The solution of

However, if equation is divided by

and sum of roots is

and in

**We can follow the following geometric construction, if in #x^2+bx+c=0#, #c<0# and signs of roots are opposite. If #b<0# smaller root is negative (and larger root is positive) and if #b>0#, larger root is negative ((and larger root is negative)).**

In such cases, we **draw a circle of diameter equal to difference of roots** (i.e. **equal to square-root of product of roots** (i.e. **and** **represent the two roots of the equation** .

So when draw a circle of diameter

**If in #x^2+bx+c=0#, #c>0#, we can follow the following geometric construction. In this case, if #b<0# both roots are positive and if #b>0#, both roots are negative.**

As in such cases, sum of roots and product of roots are known, **one can draw a circle with diameter equal to sum of the roots i.e.** **draw a line parallel to its diameter at a distance equal to square-root of product of roots i.e.**

In case any of these construction is not possible, we may have determinant