For what value of `x,x^(2) - 4x -5 and x ^(3) - 4x ^(2) - 7x + 10` eliminate ?

Options

A. 4
B. 3
C. 5
D. None of these

Answer

To solve the problem of finding the value of \( x \) for which the expressions \( x^2 - 4x - 5 \) and \( x^3 - 4x^2 - 7x + 10 \) eliminate (i.e., both equal zero), we will follow these steps:

### Step 1: Factor the Quadratic Expression
We start with the quadratic expression:
\[ x^2 - 4x - 5 \]
To factor this, we look for two numbers that multiply to \(-5\) (the constant term) and add to \(-4\) (the coefficient of \(x\)). The numbers \(-5\) and \(1\) satisfy these conditions.

Thus, we can factor the quadratic as:
\[ (x - 5)(x + 1) = 0 \]

### Step 2: Find Roots of the Quadratic
Setting the factors equal to zero gives us:
1. \( x - 5 = 0 \) → \( x = 5 \)
2. \( x + 1 = 0 \) → \( x = -1 \)

So, the roots of the quadratic are \( x = 5 \) and \( x = -1 \).

### Step 3: Substitute Roots into the Cubic Expression
Next, we need to check if these values also make the cubic expression \( x^3 - 4x^2 - 7x + 10 \) equal to zero.

#### Checking \( x = 5 \):
Substituting \( x = 5 \) into the cubic:
\[ 5^3 - 4(5^2) - 7(5) + 10 = 125 - 100 - 35 + 10 = 0 \]
Thus, \( x = 5 \) is a root of the cubic expression.

#### Checking \( x = -1 \):
Substituting \( x = -1 \) into the cubic:
\[ (-1)^3 - 4(-1)^2 - 7(-1) + 10 = -1 - 4 + 7 + 10 = 12 \neq 0 \]
Thus, \( x = -1 \) is not a root of the cubic expression.

### Conclusion
The only value of \( x \) that eliminates both expressions is:
\[ \boxed{5} \]