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

Step-by-step solution

1. 1

   Identify the coefficients carefully. The middle term is $-x$, so $b=-1$, not $1$:

   $a=4,\qquad b=-1,\qquad c=-2$

2. 2

   Try the AC method first and see it fail. We would need integers with product $ac=-8$ and sum $-1$: the pairs are $(1,-8),(2,-4),(4,-2),(8,-1)$ with sums $-7,-2,2,7$. None gives $-1$, so there is no integer factorisation and the formula is the right tool.

3. 3

   Compute the discriminant.

   $\Delta=(-1)^2-4(4)(-2)=1+32=33$

4. 4

   Apply the quadratic formula. Note $-b=-(-1)=+1$:

   $x=\frac{1\pm\sqrt{33}}{8}$

   Since $33=3\cdot 11$ contains no square factor, $\sqrt{33}$ is already in simplest form and nothing cancels with the $8$.

5. 5

   Check numerically. $\sqrt{33}\approx 5.7446$, so $x\approx 0.8431$ or $x\approx-0.5931$. Then $4(0.8431)^2-0.8431-2\approx 2.843-2.843\approx 0$ $\checkmark$. Vieta: the roots sum to $\tfrac{1}{4}$ and multiply to $-\tfrac{1}{2}$, matching $-\tfrac{b}{a}$ and $\tfrac{c}{a}$.