Question 245864
The quadratic formula is:
{{{x = (-b +- sqrt(b^2 -4ac))/2a}}}
The expression in the square root, {{{b^2 - 4ac}}}, is called the discrimnant, because its value can be used to discriminate between the different types of solutions that are possible:<ul><li>{{{b^2 - 4ac > 0}}} results in two real solutions<ul><li>If {{{b^2 - 4ac}}} is a perfect square (like 4, 9 64, 100, etc.) then you get two rational solutions.</li><li>If {{{b^2 - 4ac}}} is a not perfect square then you get two irrational solutions.</li></ul></li><li>If {{{b^2 - 4ac = 0}}} then you get a single real (and rational since 0 is a perfect square) solution.</li><li>If {{{b^2 - 4ac < 0}}} then you get two complex solutions. (Note: you only get imaginary solutions if the discriminant is negative <b>and</b> b = 0!)</li></ul>
The logic behind all of this is:<ul><li>Only zero has a single square root. Any other number will have two square roots, one positive and one negative.</li><li>The square roots of positive numbers are real (either rational or irrational).</li><li>The square roots of negative numbers are imaginary. And these imaginary roots, combined with the real number, the -b, in the numerator of the quadratic formula make the solutions complex (unless b = 0 in which case the solutions are pure imaginary numbers).</li></ul>
The discriminant of your equation
{{{25x^2 - 10x + 1 = 0}}}
is
{{{(-10)^2 - 4(25)(1)}}}
which simplifies to
{{{100 - 4(25)(1)}}}
{{{100 - 100}}}
{{{0}}}
Since the discriminant of {{{25x^2 - 10x + 1 = 0}}} is 0, there will be a single rational root.