Solve the inequality x² + x - 1 is greater than or equal to 0.

Step-by-step solution

1. 1

   Locate the roots with the quadratic formula. With $a=1$, $b=1$, $c=-1$,

   $\Delta=1^2-4(1)(-1)=1+4=5\quad\Rightarrow\quad x=\frac{-1\pm\sqrt5}{2}$

   No integer pair multiplies to $-1$ and adds to $1$, so factoring was never going to work.

2. 2

   Recognise the numbers. $\frac{-1+\sqrt5}{2}=0.6180340$ and $\frac{-1-\sqrt5}{2}=-1.6180340$ — these are the negatives of the golden ratio and its conjugate. The golden ratio satisfies $\varphi^2=\varphi+1$; substituting φ = −x turns that relation into exactly $x^2+x-1=0$. Their product is $-1$ and their sum is $-1$, matching $c/a$ and $-b/a$.

3. 3

   Apply the opening direction. The leading coefficient is positive, so the parabola opens upward and the expression is $\ge 0$ outside the interval between the roots:

   $x\le\frac{-1-\sqrt5}{2}\qquad\text{or}\qquad x\ge\frac{-1+\sqrt5}{2}$

4. 4

   Include the endpoints. The inequality is non-strict ($\ge$), and at each root the expression equals exactly $0$, so both boundary values belong to the solution set — closed brackets, not open ones.

5. 5

   Verify with test points. At $x=-2$: $4-2-1=1\ge 0$ ✓. At $x=0$: $-1$, which fails, correctly excluding the middle ✓. At $x=1$: $1+1-1=1\ge 0$ ✓. So the solution is

   $\left(-\infty,\tfrac{-1-\sqrt5}{2}\right]\cup\left[\tfrac{-1+\sqrt5}{2},\infty\right)$