# Quadratics & Methods of Solving

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.