# Segments Formed by Two Secants

How to find the segments formed by two intersecting secants of a circle: formula, 1 example, and its solution.

## Formula

PB and PD are the secants

that start from the same point P.

Then four segments are formed:

PA, AB, PC, and CD.

Then

PA⋅PB = PC⋅PD.

PA⋅(PA + AB) = PC⋅(PC + CD)

## Example

x, 3, 4, and 6

are the segments

formed by two intersecting secants.

So x(x + 3) = 4(4 + 6).

x(x + 3) = x^{2} + 3x

Multiply a Monomial and a Polynomial

4 + 6 = 10

4⋅10 = 40

Move 40 to the left side.

Factor the right side

x^{2} + 3x - 40.

Find a pair of numbers

whose product is the constant term -40

and whose sum is the coefficient of the middle term +3.

-5⋅8 = -40

-5 + 8 = +3

Then (x - 5)(x + 8) = 0.

Factor a Quadratic Trinomial

Case 2) x + 8 = 0

Then x = -8.

x is a segment.

So x cannot be minus.

So x = -8 cannot be the answer.

Case 1) x = 5

Case 2) No root

So x = 5.

So x = 5 is the answer.