How can you tell when a function has irrational or rational solutions?

Question

i have the equation 2x^2-14x=-x

how can i tell if its solutions are rational or irrational??

Answer 1

The discriminant identifies the number and type of solutions to any quadratic equation. The discriminant is the part of the quadratic formula ((-b±√(b² -4ac))÷2a where ax² + bx + c = 0) under the square root: So the value of b²-4ac determines how many and what type of solutions there are to any quadratic equation.

If b² -4ac = 0 then there is one rational solution

If the square root of b² -4ac > 0 and is an integer then there are two rational solutions

If the square root of b² -4ac > 0 and is not an integer then there are two irrational solutions

If b² -4ac < 0 then there are two complex (neither rational nor irrational) solutions

b^2-4ac = (-13)^2-4(2)(0) = 169 The square root of 169 = 13 so there are two rational solutions. Using the quadratic formula the solutions are (13-13)÷2(2) =0 and (13+13)÷2(2)= 61/2.

Answer 2

2x² + x - 14 = 0

Examine b² - 2ac from the Quadratic Formula

1 + 56

57

The square root of 57 is irrational.