Question

I'm have problems solving this equations can someone please help me

Answer 1

Well, like many problems, I think it would be best to start with writing the information we already know. First, of all we know

• $f(5) = 0$
• $f(-2) = 70$
• $f(x) = 6x^3 - 23x^2 + cx + 100$

Using this information, we can create the following equations:

• $f(5) = 6(5)^3 - 23(5)^2 + 5x + 100 = 0$
• $f(-2) = 6(-2)^3 - 23(-2)^2 - 2c + 100 = 70$

Simplifying, we get the following:

• $275 + 5c = 0$
• $-40 - 2c = 70$

Now, let's solve one of the equations for $c$:

$275 + 5c = 0$

$275 = -5c$

$c = -55$

Now, we can substitute in 55 for $c$ to get the following:

$f(x) = 6x^3 - 23x^2 - 55x + 100$

Now we need to solve $f(x)$. We already know that $x = 5$ is a zero of the equation, so let's divide our entire equation by $(x - 5)$ using synthetic division. This is a little hard to show here, but our result is:

$6x^2 + 7x - 20$

Now, we can solve by factoring:

$(3x - 4)(2x + 5) = 0$

$3x - 4 = 0$

$x = \dfrac{4}{3}$

$2x + 5 = 0$

$x = - \dfrac{5}{2}$

Our final zeros are $\boxed{x = 5, \dfrac{4}{3}, - \dfrac{2}{5}}$