Demonstrate that (5 - i√5)/5 is a solution to the equation 5x² - 10x + 6 = 0 by substituting it in.

To show by substitution that (5-i√5)/5 is a solution to the quadratic equation 5x²-10x+6=0, we will plug the value (5-i√5)/5 into the equation in place of x and see if the equation holds true.

Let x = (5-i√5)/5.

Now, we substitute x into the quadratic equation:

5x² - 10x + 6 = 0

First, calculate x²:

x² = [(5-i√5)/5]² x² = (5 - i√5)² / 5² x² = (25 - 2*5*i√5 + (i√5)²) / 25 x² = (25 - 10i√5 - 5) / 25 (because i² = -1) x² = (20 - 10i√5) / 25

Now substitute x² and x into the original equation:

5[(20 - 10i√5) / 25] - 10[(5-i√5)/5] + 6 = 0

Simplify:

(5 * 20 / 25) - (5 * 10i√5 / 25) - (10 * 5/5) + (10 * i√5/5) + 6 = 0 (4) - (2i√5) - (10) + (2i√5) + 6 = 0

Now you'll notice that -2i√5 and +2i√5 cancel each other out:

4 - 10 + 6 = 0 -6 + 6 = 0 0 = 0

Since the left and right sides of the equation are equal, this verifies that x = (5-i√5)/5 is indeed a solution to the quadratic equation 5x²-10x+6=0.