Exercise 7 Page 520

To factor a quadratic expression of the form ax^2+bx+c, we need to follow three steps.

1. Find two integers that multiply to ac and add to b.
2. Rewrite the linear term as the sum of two linear terms whose coefficients are the numbers found in the previous step.
3. Factor by groups the resulting expression.

Let's consider what happens in the first step if a=1. ac ⇒ 1c=cIf a=1, then the first step simplifies to finding two integers that multiply to c and add to b. Steps 2 and 3 stay the same. Let's consider one example in which a=1, and one example in which a≠ 1.

a=1

Let's factor the quadratic expression x^2+2x-3. We will start by identifying the values of a, b, and c. x^2+2x-3 ⇔ 1x^2+ 2x+( - 3) In this case, we have that a = 1, b = 2, and that c = - 3. Therefore, we have to find two integers whose product is - 3 and whose sum is 2. These integers are 3 and - 1. 3(- 1)= - 3 and 3+(- 1)= 2 We can then rewrite the linear term and factor the groups.

a≠ 1

Now, let's factor the quadratic expression 2x^2+4x-6. Just like in the previous example, we will start by identifying the values of a, b, and c. 2x^2+4x-6 ⇔ 2x^2+ 4x+( - 6) In this case, we have that a = 2, b = 4, and that c = - 6. Therefore, we have to find two integers whose product is 2( - 6)=- 12 and whose sum is 4. These integers are 6 and - 2. 6(- 2)=- 12 and 6+(- 2)=4 We can then rewrite the linear term and factor by groups.