What are the excluded values of $x$ for $\frac{x+4}{-3 x^2+12 x+36}$?
To find the excluded values for the rational expression \(\frac{x+4}{-3x^2+12x+36}\), we must identify the values of \(x\) that make the denominator equal to zero. Division by zero is undefined in mathematics, so any \(x\) that results in a zero denominator is excluded from the domain of the expression. The denominator is given by the quadratic expression: \(-3x^2+12x+36\). Therefore, we need to solve the equation \(-3x^2+12x+36 = 0\).
We start by simplifying the equation \(-3x^2+12x+36 = 0\). Notice that each term is divisible by \(-3\). Dividing both sides of the equation by \(-3\), we obtain:
\(\frac{-3x^2}{-3} + \frac{12x}{-3} + \frac{36}{-3} = \frac{0}{-3}\)
Which simplifies to:
\(x^2 - 4x - 12 = 0\)
Now we solve the quadratic equation \(x^2 - 4x - 12 = 0\) by factoring. We look for two numbers that multiply to \(-12\) and add up to \(-4\). These numbers are \(-6\) and \(2\), since \((-6) \times 2 = -12\) and \((-6) + 2 = -4\).
Thus, the expression factors as:
\((x - 6)(x + 2) = 0\)
Setting each factor equal to zero gives the solutions:
\(x - 6 = 0 \implies x = 6\)
\(x + 2 = 0 \implies x = -2\)
Therefore, the excluded values are \(x = 6\) and \(x = -2\).
Excluded values are crucial in many fields like engineering and physics when working with transfer functions or modeling systems where certain inputs could cause a system to crash or behave uncontrollably. For example, if a model's stability is represented by a rational function, the excluded values tell you which specific input conditions will lead to an "undefined" or catastrophic state, allowing engineers to avoid those points when designing safe operating parameters for machines or electrical circuits.