Solve 2x² − x − 15 = 0.

Step-by-step solution

1. 1

   Set up the ac method. With $a=2$, $b=-1$, $c=-15$, look for two numbers whose product is $ac=2(-15)=-30$ and whose sum is $b=-1$. Because $a\neq 1$, factoring $c$ alone would not work.

2. 2

   Find the pair. Factor pairs of $-30$ include $(1,-30),(2,-15),(3,-10),(5,-6),(6,-5),(10,-3)$. The pair summing to $-1$ is $5$ and $-6$.

3. 3

   Split the middle term and group.

   $2x^{2}+5x-6x-15=0$

   $x(2x+5)-3(2x+5)=0$

   $(2x+5)(x-3)=0$

   The repeated bracket $(2x+5)$ confirms the split was correct.

4. 4

   Apply the zero-product property.

   $2x+5=0\Rightarrow x=-\frac52,\qquad x-3=0\Rightarrow x=3$

   $\boxed{x=3\ \text{or}\ x=-\tfrac52}$

5. 5

   Check both roots. At $x=3$: $2(9)-3-15=18-18=0$ ✓. At $x=-\tfrac52$: $2\left(\tfrac{25}{4}\right)+\tfrac52-15=\tfrac{25}{2}+\tfrac52-15=15-15=0$ ✓.

6. 6

   Confirm with Vieta and with the discriminant. The roots should sum to $-b/a=\tfrac12$ and multiply to $c/a=-\tfrac{15}{2}$: $3-\tfrac52=\tfrac12$ ✓ and $3\left(-\tfrac52\right)=-\tfrac{15}{2}$ ✓. The discriminant $1+120=121=11^{2}$ is a perfect square, which is why the quadratic factored over the rationals in the first place.