The equation x^2 + k = 6 x has two distinct real roots. Find the range of values of k.

I know that you do
36 -4k > 0
-4k > -36
k < 9
But I don't get why you switch the signs. So. When do u switch signs

Switching dircetion is just a shorthand way of moving stuff from one side to the other:

-4k > -36
add 4k to both sides:
0 > 4k-36
add 36 to both sides:
36 > 4k
or, as you saw above,
4k < 36
k < 9

multiplying and dividing by a negative value change the direction, just as moving stuff from one side to the other reverses the sense of the inequality.

When solving an inequality, you switch the sign when you multiply or divide both sides of the inequality by a negative number. This is because multiplying or dividing both sides by a negative number will reverse the inequality.

In the given equation, x^2 + k = 6x, we want to find the range of values of k for which the equation has two distinct real roots. To do this, we can use the discriminant of the quadratic equation.

The discriminant is defined as b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. In this case, a = 1 (coefficient of x^2), b = -6 (coefficient of x), and c = k.

For the equation to have two distinct real roots, the discriminant must be greater than zero (D > 0). In other words, we need to solve the inequality D > 0.

For our equation, the discriminant is calculated as follows:
D = (-6)^2 - 4(1)(k)
= 36 - 4k

We want to find the range of values of k such that D > 0:
36 - 4k > 0

To solve this inequality, we can start by isolating k:
-4k > -36

Now, to find the range of values for k, we divide both sides of the inequality by -4. However, remember the rule about switching the sign when dividing by a negative number. Since we are dividing by -4, we need to flip the inequality sign:
k < 9

Therefore, the range of values for k that satisfies the condition for the equation to have two distinct real roots is k < 9.