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.