Question

Given `g\left(x\right)=x^{2}-3x-10`, which statement is true?
the zeros are what? because the factors are what?

Answer 1

To find the zeros of the function \( g(x) = x^2 - 3x - 10 \), we need to solve the equation \( g(x) = 0 \):

\[
x^2 - 3x - 10 = 0
\]

We can factor this quadratic equation. We are looking for two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of the linear term). The numbers that satisfy these conditions are \(-5\) and \(2\).

Thus, we can express the quadratic as:

\[
g(x) = (x - 5)(x + 2)
\]

Now, to find the zeros, we set each factor equal to zero:

1. \( x - 5 = 0 \) \(\Rightarrow x = 5\)
2. \( x + 2 = 0 \) \(\Rightarrow x = -2\)

So, the zeros of \( g(x) \) are \( x = 5 \) and \( x = -2 \).

The factors of the quadratic are \( (x - 5) \) and \( (x + 2) \).

In summary:
- The zeros are \( 5 \) and \( -2 \).
- The factors are \( (x - 5) \) and \( (x + 2) \).