Solve by factoring, give all the real solutions:
$-5x^3 + 15x^2 + 50x = 0$
The given equation is \(-5x^3 + 15x^2 + 50x = 0\). We are asked to find all real solutions for \(x\) by factoring. This is a cubic polynomial equation, and we can start by looking for common factors among the terms.
Observing the coefficients \(-5, 15, 50\), we see that each term is divisible by \(-5\). Furthermore, each term contains the variable \(x\). Therefore, we can factor out the greatest common factor, which is \(-5x\). Factoring \(-5x\) from the equation gives:
\[-5x(x^2 - 3x - 10) = 0\]
Now we focus on the quadratic expression \(x^2 - 3x - 10\). We seek two numbers that multiply to \(-10\) and add up to \(-3\). The factors of \(-10\) are \((1, -10), (-1, 10), (2, -5),\) and \((-2, 5)\). Adding these pairs, we find \(2 + (-5) = -3\). Thus, the quadratic factors as \((x - 5)(x + 2)\). Substituting this back into our equation, we obtain:
\[-5x(x - 5)(x + 2) = 0\]
According to the Zero Product Property, if a product of factors is equal to zero, then at least one of the factors must be zero. Setting each factor equal to zero:
\(-5x = 0 \implies x = 0\)
\(x - 5 = 0 \implies x = 5\)
\(x + 2 = 0 \implies x = -2\)
Thus, the real solutions to the equation are \(x = 0, 5, -2\).
Factoring is essential in many real-world applications, such as physics and engineering, where we often need to find the "roots" or equilibrium points of a system. For example, if the height of a projectile launched from a cliff is described by the function \(h(t) = -5t^3 + 15t^2 + 50t\), solving \(h(t) = 0\) helps determine at what times the projectile hits the ground. By factoring, we can quickly identify these critical moments in time, allowing us to predict the trajectory and behavior of physical systems accurately.