Graphical Solution of a Quadratic System

Graphical solution of a quadratic system

A system of two quadratic equations, or of a quadratic and a linear equation, may be solved by a method like the one used for two linear equations: graph both equations on the same axes and to the same scale. Each point of intersection of the two graphs determines one set of roots.

Example 1. Solve graphically \begin{cases} x^2 + y^2 = 34 \\ x + 2y + 7 = 0 \end{cases}

Constructing the graphs of the two equations on the same axes, we obtain the circle and the straight line of the figure. For A, (x, y) = (3, -5); for B, (x, y) = (-5\tfrac45, -\tfrac35).

Example 2. Solve graphically \begin{cases} 3x - 4y = 8 \\ x^2 = 4y + 18 \end{cases}

The graphs are a straight line and a parabola. There are two sets of roots, corresponding to the two points of intersection: A, x = 5, y = 1\tfrac34; and B, x = -2, y = -3\tfrac12.

If the straight line were moved downward, always parallel to its present position, the points A and B would approach each other and finally coincide. Then the line would be tangent to the parabola, at the point (\tfrac32, -3\tfrac{15}{16}). Were the line moved still farther, it would neither touch nor intersect the parabola, and there would be no graphical solution.

Example 3. Solve graphically \begin{cases} x^2 - xy + y^2 = 25 \\ x^2 - y^2 = 8 \end{cases}

The graphs are an ellipse and a hyperbola. There are four sets of roots, one for each point of intersection, approximately A\,(-3.6,\ 2.2), B\,(5.4,\ 4.5), C\,(3.6,\ -2.2), D\,(-5.4,\ -4.5). (The book reads 4.7 for the y of B and D; a more careful graph gives 4.5.)

A smaller number in place of 25 would give a smaller ellipse; one number would make the ellipse just touch the hyperbola, and a still smaller one would give an ellipse that neither touches nor intersects it. These conditions give (a) four sets of real roots, (b) two sets of real roots, (c) no set of real roots. You can solve such a system graphically, though you will scarcely be able to solve it algebraically even after mastering this course.

Examples 1, 2, and 3 partially illustrate this statement: if, in a system of two equations in two variables, one equation is of the mth degree and one of the nth, there are usually mn sets of roots (real or imaginary), and never more than mn such sets.

Typing the answers

Give each set of roots as a pair (x, y), separated by commas: (3, -5), (-5.8, -0.6). The order of the pairs does not matter, but each pair must be in the order (x, y). Readings from a graph are accepted within about 0.2; exact values are always accepted. If the graphs do not meet, type none.

Check with the calculator. The calculator's Graph tab draws these curves directly: type the equation as it stands, for example x^2 + y^2 = 25 or xy = 6, or press the Graph button beside an equation in the lesson. Draw your own graph first; then compare.

Try one: