Question:

Consider the following sequence of steps to solve the equation: -15 + 4x + 4x2 = 0

Step 1 ⇒ 4x2 + 4x - 15 = 0
Step 2 ⇒ 4x2 - 6x + 10x - 15 = 0
Step 3 ⇒ 2x(2x - 3) + 5(2x - 3) = 0
Step 4 ⇒ (2x + 5)(2x - 3) = 0
Step 5 ⇒ x = -3/2 or x = 5/2

A. Explain why Step 1 is valid using properties of the real number system.
Show possible intermediate steps to support your explanation.

A. Explain why Step 4 is valid.

Step 1:

Addition is commutative

A + B = B + A

So the terms can be rearranged

Step 4:

2x(2x - 3) + 5(2x - 3)

Since (2x - 3) is common to both terms,

You can take it common

(2x - 3)(2x + 5)

This can be proved verified using distribution property

(2x - 3)(2x + 5) is the same as

(2x + 5)(2x - 3) because multiplication is commutative

m∠5 + m∠3 = m∠1   is true.

m∠5 + m∠6 = 180 is true

m∠2 +m∠3 = m∠6   is true

m∠2 +m∠3 +m∠5 = 180   is true