What is the radix of the numbers if the solution to the quadratic equation $$x^2 -10x + 26 = 0$$​ is x = 4 and x = 7?

To find the radix (also known as the sum of the roots) of the quadratic equation $$x^2 - 10x + 26 = 0$$​ given that the solutions are x = 4 and x = 7, follow the steps below:

We are given that the solutions to the quadratic equation are x = 4 and x = 7.

By Vieta's formulas, for a quadratic equation of the form $$ax^2 + bx + c =0$$​, the sum and product of the roots are:

The radix is defined as the sum of the roots of the equation.

Given roots $$x_1 = 4$$​ and $$x_2 = 7$$​:

​$$\text{Radix} = x_1 + x_2 = 4 + 7 = 11$$​​

To double-check, we substitute one of the solutions, say x = 4, into the equation and convert the coefficients to base 10.

The quadratic equation is:

​$$x^2 - 10x + 26 = 0$$​​

Substituting x = 4:

​$$(4)^2 - 10(4) + 26 = 0$$​​

​$$16 - 40 + 26 = 0$$​​

​$$2r = 22$$​​

From the above equation, we can solve for r:

​$$r = 11$$​​

Thus, the radix is confirmed as 11.