Solve the equation x^2 + 4x - 21 = 0.

Step-by-step solution

1. 1

   Read the signs before hunting for factors. The constant term is negative, which forces the two factors to have opposite signs. The middle coefficient is positive, so the larger factor in absolute value must be the positive one.

2. 2

   List the factor pairs of $21$ and pick the one that works. The candidates are $1\cdot 21$ and $3\cdot 7$. Assigning opposite signs, we need a difference of $4$:

   $7+(-3)=4,\qquad 7\cdot(-3)=-21$

   so the pair is $7$ and $-3$.

3. 3

   Write the factorisation.

   $x^2+4x-21=(x+7)(x-3)=0$

4. 4

   Solve each factor.

   $x+7=0\ \Rightarrow\ x=-7,\qquad x-3=0\ \Rightarrow\ x=3$

5. 5

   Check both roots. At $x=-7$: $49-28-21=0$ $\checkmark$. At $x=3$: $9+12-21=0$ $\checkmark$. Note the sign flip between factor and root: the factor $(x+7)$ gives the root $-7$, not $+7$.