1. solve w2 + 13w + 42 = 0 by factoring.

Step-1 : Multiply the coefficient of the first term by the constant 1 • 42 = 42

Step-2 : Find two factors of 42 whose sum equals the coefficient of the middle term, which is -13 .

-42 + -1 = -43
-21 + -2 = -23
-14 + -3 = -17
-7 + -6 = -13 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -7 and -6
w2 - 7w - 6w - 42

Step-4 : Add up the first 2 terms, pulling out like factors :
w • (w-7)
Add up the last 2 terms, pulling out common factors :
6 • (w-7)
Step-5 : Add up the four terms of step 4 :
(w-6) • (w-7)
Which is the desired factorization

Equation at the end of step 1 :

(w - 6) • (w - 7) = 0
Step 2 :

Theory - Roots of a product :

2.1 A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

2.2 Solve : w-6 = 0

Add 6 to both sides of the equation :
w = 6

Solving a Single Variable Equation :

2.3 Solve : w-7 = 0

Add 7 to both sides of the equation :
w = 7

Supplement : Solving Quadratic Equation Directly

Solving w2-13w+42 = 0 directly
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula W=7,w=6

Really any number between them works. for example 144.951