Compare using the quadratic formula to find solutions to a quadratic equation having irrational roots to that of one that has rational roots.

For a general quadratic equation with rational coefficients

the two solutions are:

where D is the determinant.

Clearly, a solution will a rational number as long as is rational. However, it can be shown that a square root of an integer is only rational if its value is an integer. In other words, is rational if and only if the determinant is a perfect square, , otherwise the square root is irrational. Therefore the coefficients of quadratic equations that are to have rational solutions must satisfy the following condition:

is means division by fractions

step-by-step explanation:

27/4x^6 y^8

step-by-step explanation:

4(3x^2y^4)^3 (2x^3y^5)^4

=

108x^6 y^12 /16x^12 y^20

=

108 /16x^6 y^8

=

27 /4x^6 y^8