Solve 2x² − x − 6 = 0.

Step-by-step solution

1. 1

   Use the ac method because the leading coefficient is not 1. With $a=2$, $b=-1$, $c=-6$, you look for two numbers whose product is $ac=2\times(-6)=-12$ and whose sum is $b=-1$. Guessing factor pairs of $c$ alone would fail here.

2. 2

   Find the pair. The factor pairs of $-12$ are $(1,-12),(2,-6),(3,-4),(4,-3),(6,-2),(12,-1)$; the pair summing to $-1$ is $3$ and $-4$, since $3+(-4)=-1$ and $3\times(-4)=-12$.

3. 3

   Split the middle term and group.

   $2x^2+3x-4x-6=0$

   $x(2x+3)-2(2x+3)=0$

   $(2x+3)(x-2)=0$

   The common bracket $(2x+3)$ appearing in both groups confirms the split was done correctly.

4. 4

   Apply the zero-product property.

   $2x+3=0\Rightarrow x=-\frac32,\qquad x-2=0\Rightarrow x=2$

   $\boxed{x=2\ \text{or}\ x=-\tfrac{3}{2}}$

5. 5

   Check both roots. At $x=2$: $2(4)-2-6=8-8=0$. At $x=-\tfrac32$: $2\left(\tfrac94\right)+\tfrac32-6=\tfrac92+\tfrac32-6=6-6=0$. Both satisfy the equation.

6. 6

   Confirm with Vieta's formulas. The roots should sum to $-b/a=\tfrac12$ and multiply to $c/a=-3$. Indeed $2+(-\tfrac32)=\tfrac12$ and $2\times(-\tfrac32)=-3$, so no root was mis-signed.