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:

b² - 8b - 19 = -3b + 5
b² - 8b - 19 - 5 = -3b
b² - 8b + 3b - 24 = 0
b² - 5b - 24 = 0

Next, factor the quadratic equation:

b² - 5b - 24 = 0
(b + 3)(b - 8) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (b + 3) or (b - 8) must equal zero:

If (b + 3) = 0, b must equal -3
If (b - 8) = 0, b must equal 8

So the solution is that b = -3 or 8

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, \(c = \sqrt{a + b}\)

A right angle measures 90 degrees and is the intersection of two perpendicular lines. In diagrams, a right angle is indicated by a small box completing a square with the perpendicular lines.

To find the diagonal we need to know the length of one of the square's sides. We know the area and the area of a square is the length of one side squared:

a = s²

so the length of one side of the square is:

s = \( \sqrt{a} \) = \( \sqrt{9} \) = 3

The Pythagorean theorem defines the square of the hypotenuse (diagonal) of a triangle with a right angle as the sum of the squares of the other two sides:

c² = a² + b²
c² = 3² + 3²
c² = 18
c = \( \sqrt{18} \) = \( \sqrt{9 x 2} \) = \( \sqrt{9} \) \( \sqrt{2} \)
c = 3\( \sqrt{2} \)

A parallelogram is a quadrilateral with two sets of parallel sides. Opposite sides (a = c, b = d) and angles (red = red, blue = blue) are equal. The area of a parallelogram is base x height and the perimeter is the sum of the lengths of all sides (a + b + c + d).