In details

Composition of the equation


Every real rooted square equation x1, x2, x3 and x4 can be composed by the formula:

(x -x1). (x - x2). (x - x3). (x - x4) = 0

Example:

  • Compose the square equation whose roots are:

    Solution:

    The) (x - 0) (x - 0) (x + 7) (x - 7) = 0
    x2(x2 -49) = 0
    x4 - 49x2 = 0
    B) (x + a) (x - a) (x + b) (x - b) = 0
    (x2-The2) (x2-B2) = 0
    x4 - (a2 + b2) x2 + a2B2 = 0

Root properties of the equation

Consider the equation ax4 + bx2 + c = 0, whose roots are x1, x2, x3 and x4 and the equation of the 2nd degree ay2 + by + c = 0, whose roots are y 'and y ". From each root of the 2nd degree equation, we get two symmetrical roots for the square. So:

From the above, we can establish the following properties:

1st property: the sum of the real roots of the equation is zero.

x1 + x2 + x3 + x4 = 0

2nd property: the sum of the squares of the real roots of the square equation equals -.

3rd property: the product of the real and nonzero roots of the equation equals .

Next: Irrational Equations