Given `g\left(x\right)=x^{2}-3x-10`, which statement is true?

Question

the zeros are 5 and -2, because the factors are x- 5 and x + 2
the zeros are -5 and -2 because the factors are x + 5 and x + 2

the zeros are 2 and - 5, because the factors are x - 2 and x + 5

the zeros are 2 and 5, because the factors are x - 2 and x - 5

Answer 1

To find the zeros of the function \( g(x) = x^2 - 3x - 10 \), we need to set the function equal to zero and solve for \( x \):

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

Next, we can factor the quadratic. 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 \(-5\) and \(2\) fit this condition:

\[ x^2 - 5x + 2x - 10 = (x - 5)(x + 2) = 0 \]

Thus, the factors are \( (x - 5) \) and \( (x + 2) \). Setting each factor to zero gives us the zeros:

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

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

Based on the stated options, the correct statement is:

the zeros are 5 and -2, because the factors are x - 5 and x + 2.

Answer 2

What is the solution set for`x^{2}-x-72=0`?

Answer 3

To solve the equation \( x^2 - x - 72 = 0 \), we will first attempt to factor it. We are looking for two numbers that multiply to \(-72\) (the constant term) and add to \(-1\) (the coefficient of the linear term).

The pairs of factors of \(-72\) that we can consider are:

• \(1\) and \(-72\)
• \(-1\) and \(72\)
• \(2\) and \(-36\)
• \(-2\) and \(36\)
• \(3\) and \(-24\)
• \(-3\) and \(24\)
• \(4\) and \(-18\)
• \(-4\) and \(18\)
• \(6\) and \(-12\)
• \(-6\) and \(12\)
• \(8\) and \(-9\)
• \(-8\) and \(9\)

We find that the numbers \(8\) and \(-9\) multiply to \(-72\) and add up to \(-1\).

Thus, we can factor the quadratic as follows:

\[ x^2 - x - 72 = (x - 9)(x + 8) = 0 \]

Next, we set each factor equal to zero:

1. \( x - 9 = 0 \) → \( x = 9 \)
2. \( x + 8 = 0 \) → \( x = -8 \)

The solution set for the equation \( x^2 - x - 72 = 0 \) is:

\[ { 9, -8 } \]