# 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)?

Feb 21, 2017

#### Explanation:

The solution of $a {x}^{2} + b x + c = 0$ is given by $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

However, if equation is divided by $a$, it becomes of the form ${x}^{2} + b x + c = 0$, whose solution is $x = \frac{- b \pm \sqrt{{b}^{2} - 4 c}}{2}$

and sum of roots is $- b$ and product is $c$,

and in ${x}^{2} + 4 x - 32 = 0$, product of roots would be $32$ and as its sign is negative, difference of roots is $4$, with smaller root being positive and larger root being negative.

We can follow the following geometric construction, if in ${x}^{2} + b x + c = 0$, $c < 0$ 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. $b$) and then a tangent $C P$ equal to square-root of product of roots (i.e. $\sqrt{c}$) and then join $P$ to the center of circle as shown, then $B P$ and $D P$ represent the two roots of the equation .

So when draw a circle of diameter $4$ units and draw a tangent of length equal to $\sqrt{32}$ and join as shown in figure, we will find $B P$ and $D P$ as $4$ and $8$ respectively and hence roots are $4$ and $- 8$. If in ${x}^{2} + b x + c = 0$, $c > 0$, 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. $b$, ($A B$ in figure below) and then draw a line parallel to its diameter at a distance equal to square-root of product of roots i.e. $\sqrt{c}$, ($C D$ in figure below with $E F = \sqrt{c}$), cutting circle at $E$. Now draw a perpendicular $E F \bot A B$. Then $A F$ and $B F$ represent the two roots. In case any of these construction is not possible, we may have determinant ${b}^{2} - 4 a c < 0$ and roots are complex.