for the equation x^2-10x+k=0,for what value of k does the equation have a single real root?

Answer 1

A quadratic equation has but a single root iff b^2 - 4ac = 0, for if this quantity, called the Discriminant, is equal to zero, then its square root is zero and so adding or subtracting it from - b in the Quadratic Formula leads to the very same result, the single, real root.

In the case of the equation above, a is 1, b is -10, and c is k, so that b^2 - 4ac = 100 - 4k. On the condition that this equals zero, k can only be 25.

Alternative method of solution:

Recognize x^2 - 10x as part of a perfect square trinomial whose constant term is 25, or simply complete the square. Again, the value of 25 for k will make the entire equation equal zero.

Answer 2

Since all quadratic equations have two roots and imaginary roots have to come in pairs, the only way this can be true is if the equation has one root with a multiplicity of two. That means it takes the factored form:

(x-c)(x-c)=0, and c^2 is equal to our constant k.