Solve the equation x squared minus x minus 42 equals 0 by factoring.

Step-by-step solution

1. 1

   Set up the search. For a monic quadratic $x^2+bx+c$ you need two numbers whose product is $c$ and whose sum is $b$. Here

   $\text{product}=-42,\qquad \text{sum}=-1$

2. 2

   Use the signs to narrow the search fast. A negative product means the two numbers have opposite signs; a sum of $-1$ that is small compared with $42$ means they are close together in size, with the negative one slightly larger. The factor pairs of $42$ are $1\cdot 42$, $2\cdot 21$, $3\cdot 14$, $6\cdot 7$ — and only $6$ and $7$ differ by $1$.

3. 3

   Fix the signs and write the factors. Taking $-7$ and $+6$ gives product $-42$ and sum $-1$ $\checkmark$:

   $x^2-x-42=(x-7)(x+6)$

4. 4

   Solve with the zero product property.

   $(x-7)(x+6)=0\ \Longrightarrow\ x=7\ \text{or}\ x=-6$

5. 5

   Check both roots in the original equation.

   $7^2-7-42=49-49=0\quad\checkmark$

   $(-6)^2-(-6)-42=36+6-42=0\quad\checkmark$