Graphical solution of quadratic equations

The zeros of the function y = x² + ax + b are the roots of the quadratic equation 0 = x² + ax + b. They are also the abscissae of the points of intersection of the parabola y=x²+ax+b with the abscissa.

Task: Solve graphically the equation x²+6x+5 = 0 .

You start with the function y = x² + 6x + 5. Hence y-5+9=x²+6x+9 or y+4=(x+3)². The vertex of the parabola has the coordinates (-3,-4) and yields, as shown, the location for your pattern of x². As a rule, it will be sufficient to place the pattern correctly on the graph paper, in order to be able to read off the abscissae of the points of intersection S₁ and S₂. The roots of the given equation are x₁ = -1 and x₂ = -5.

Solution using y =x².

So far, we had to give the parabola every time a new position. You will ask whether it is possible to manage the determination of roots, leaving y =x² in its original position. Consider the example x² + x - 2 = 0 or x² = -x - 2. If you set y = x² and y = -x + 2, then the two equations yield again for the same y the original equation. In other words, you can start from the two curves y = x² and y = -ax -b and have the parabola always in the standard position, but you must every time redraw the straight line. The line and the parabola intersect for the same values of y. The abscissae are therefore the required roots. In the figure, x₁ = +1 and x₂ = -2. Often you need not draw the line, but simply use a ruler.

Task: Find the roots of the three equations:

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