Questions: For what input values of g(x)=(x^2+7x+10)/(x^2-5x+6) are the output values of g(x)=0?

Transcript text: For what input values of $g(x)=\frac{x^{2}+7 x+10}{x^{2}-5 x+6}$ are the output values of $g(x)=0$ ?

Solution

Solution Steps

To find the input values for which $ g(x) = 0 $, we need to determine when the numerator of the function is zero, since a fraction is zero when its numerator is zero (and the denominator is not zero). Therefore, we solve the equation $ x^2 + 7x + 10 = 0 $. We also need to ensure that these solutions do not make the denominator zero by checking $ x^2 - 5x + 6 \neq 0 $.

Step 1: Solve the Numerator

To find the input values for which $ g(x) = 0 $, we set the numerator equal to zero: \[ x^2 + 7x + 10 = 0 \] Factoring the quadratic, we get: \[ (x + 5)(x + 2) = 0 \] Thus, the solutions are: \[ x = -5 \quad \text{and} \quad x = -2 \]

Step 2: Solve the Denominator

Next, we need to ensure that these solutions do not make the denominator zero. We set the denominator equal to zero: \[ x^2 - 5x + 6 = 0 \] Factoring this quadratic, we find: \[ (x - 2)(x - 3) = 0 \] The solutions are: \[ x = 2 \quad \text{and} \quad x = 3 \]

Step 3: Validate Solutions

We check if the solutions from the numerator, $ x = -5 $ and $ x = -2 $, are also solutions to the denominator. Since neither $ -5 $ nor $ -2 $ is equal to $ 2 $ or $ 3 $, both values are valid.