Quadratics & Methods of Solving
- Walter M. Stroup
ORIGINAL SYSTEM OF EQUATIONS The BLUE shows the original system of equations: x^2 - 2x + 4 = x +2 The points A & B show the intersection of these two functions or where g(x) = h(x). The x value of A is 1 and the x value of B is 2 ROOTS (& THE QUADRATIC FORMULA) The ORANGE lines show what you get when you transform the original system to get x^2 - 3x + 2 = 0. The points C & D show these two functions or where i(x) = j(x). The x value of C is 1 and the x value of D is 2. These are the values you get if you use the QUADRATIC FORMULA. Note, these are the SAME x values as for A & B. COMPLETING THE SQUARE The GREEN lines show the final step of completing the square for the expression on the LEFT once you put the constants together on the RIGHT x^2 - 3x = -2 [ - b / 2a ]^2 or [ - (-3) / 2 * 1 ] ^ 2 or 9/4 is added to both sides leaving a perfect square on the LEFT and -2 + 9/4 on the right. (x - 3/2)^2 = (-2 + 9/4) So, the final step is to set (x - 3/2) = positive (-2 + 9/4) and (x - 3/x) = negative (-2 + 9/4) OR where the line x - 3/2 o(x) intersects with positive (-2 + 9/4) [i.e., o(x) = p(x)] or o(x) intersects with negative (-2 + 9/4) [i.e., o(x) = q(x)] The points F & E show the respective intersections. The x value of is F is 1 and the x value of E is 2. These are the values you get if you COMPLETE THE SQUARE. Note, these are the SAME x values as for A & B.
You can turn on or off graphs by clicking on the "o" to the left of each of the functions. This can be helpful in making clear which graphs you are discussing (and which ones you are not).