# Segments Formed by Two Intersecting Secants

How to solve the segments formed by two intersecting secants problems: formula, example, and its solution.

## Formula

[blue]⋅[dark blue] = [green]⋅[dark green]

[blue]: Exterior segment from a secant

[dark blue]: The secant including [blue]

[green]: Exterior segment from the other secant

[dark green]: The other secant including [green]

## Example

Four segments are formed

by two intersecting secants.

The segments are [*x*, 3] and [4, 6].

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

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

4 + 6 = 10

4⋅10 = 40

Move 40 to the left side.

Then *x*^{2} + 3*x* - 40 = 0.

Factor the left side.

Factor a quadratic trinomial

Find a pair of numbers

whose product is the constant term [-40]

and whose sum is the middle term's coefficient [+3].

The constant term is (-).

So the signs of the numbers are different:

one is (+), and the other is (-).

(-1, 40), (-2, 20), and (-4, 10)

are not the right numbers.

[-40] = -5⋅8

-5 + 8 = 3 = [+3]

So -5 and 8 are the right numbers.

Use -5 and +8

to write a factored form:

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

Solve (*x* - 5)(*x* + 8) = 0.

1) *x* - 5 = 0

So *x* = 5.

2) *x* + 8 = 0

So *x* = -8.

But this cannot be the answer

because *x* should be (+).

Solving a quadratic equation by factoring

So *x* = 5 is the answer.