Explore the Quadratic Puzzle: y = x² + x - 20 Greater Than Zero

To determine the values of $x$ where the function $y = x^2 + x - 20$ is greater than zero, we follow these steps:

• First, let us find the roots of the equation $x^2 + x - 20 = 0$.
• We attempt to factor the quadratic. We need two numbers that multiply to $-20$ and add up to $1$ (the coefficient of $x$).
• The numbers $5$ and $-4$ work, since $5 \times (-4) = -20$ and $5 + (-4) = 1$.
• Thus, we can factor the quadratic expression as $(x - 4)(x + 5) = 0$.
• Setting each factor equal to zero gives us the roots $x = 4$ and $x = -5$.
• The function changes its sign at these roots.
• To determine the sign in each interval, choose test points in the intervals defined by these roots: $(-\infty, -5)$, $(-5, 4)$, and $(4, \infty)$.
• For $x < -5$, say $x = -6$: $(-6 - 4)(-6 + 5) = (-10)(-1) > 0$.
• For $-5 < x < 4$, say $x = 0$: $(0 - 4)(0 + 5) = (-4)(5) < 0$.
• For $x > 4$, say $x = 5$: $(5 - 4)(5 + 5) = (1)(10) > 0$.
• The function is positive for $x < -5$ and $x > 4$, corresponding to the intervals where $(x - 4)(x + 5) > 0$.

Therefore, the solution to the problem, for which values of $x$ make $f(x) > 0$, is $x < -5$ or $x > 4$.