A circle is a figure in which each point around its perimeter is an equal distance from the center. The radius of a circle is the distance between the center and any point along its perimeter. A chord is a line segment that connects any two points along its perimeter. The diameter of a circle is the length of a chord that passes through the center of the circle and equals twice the circle's radius (2r).

The first step to solve a quadratic expression that's not set to zero is to solve the equation so that it is set to zero:

z² + 4z - 27 = 2z - 3
z² + 4z - 27 + 3 = 2z
z² + 4z - 2z - 24 = 0
z² + 2z - 24 = 0

Next, factor the quadratic equation:

z² + 2z - 24 = 0
(z - 4)(z + 6) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (z - 4) or (z + 6) must equal zero:

If (z - 4) = 0, z must equal 4
If (z + 6) = 0, z must equal -6

So the solution is that z = 4 or -6

The entire length of this line is represented by AD which is AB + BD:

AD = AB + BD

The Pythagorean theorem defines the relationship between the side lengths of a right triangle. The length of the hypotenuse squared (c²) is equal to the sum of the two perpendicular sides squared (a² + b²): c² = a² + b² or, solved for c, \(c = \sqrt{a + b}\)

To solve this equation, repeatedly do the same thing to both sides of the equation until the variable is isolated on one side of the > sign and the answer on the other.

6c + 3 > \( \frac{c}{1} \)
1 x (6c + 3) > c
(1 x 6c) + (1 x 3) > c
6c + 3 > c
6c + 3 - c > 0
6c - c > -3
5c > -3
c > \( \frac{-3}{5} \)
c > -\(\frac{3}{5}\)