A geometrical method of solving quadratic equations.

© Stu Savory, 2006.

Question :
Solve the quadratic equation x 2 - ax + b = 0 using a ruler and compasses only.

Method :
Without loss of generality assume the coefficients a and b are both < 1.

Given : a construct with the points Y and Z having coordinates (0,1) and (a,b) respectively.

Find the centre of the circle YZ by bisecting the diameter YZ in the usual manner.

Draw the circle with diameter YZ.

The circle YZ cuts the horizontal (X) axis at X1 and X2 with coordinates (x1,0) and (x2,0) where these are the roots of the original quadratic equation x 2 - ax + b = 0 ; easy when you know how :-)

Caveat : Because this method is so simple, we need to understand why people prefer to use the algebraic method instead. That's because this method is so highly sensitive to the slightest inaccuracy in your drawing of the circle. Here are some worked examples. With the coefficient 'a' arbitrarily set to one, let us see how x1 and x2 diverge rapidly with small changes in the coefficient 'b' of the equation x 2 - ax + b = 0.

So in practice, with the point Y drawn in your exercise book at - say - 10 cm above the origin, an inaccuracy of just 1 mm in the position of Z (i.e instead of b) means that you could get x1 = x2 = 0,5 instead of x1 = 0,4 and x2 = 0,6 :-(

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