Question

What is the solution to the equation \( (x^2 - 4x - 5 = 0) \)?

• A. 1 and -5
• B. 1 and 5
• C. -1 and 5
• D. -1 and -5

Hint:

Factoring is usually the quickest method for solving quadratic equations when the numbers are simple. If you can't find the factors easily, always use the quadratic formula as it works for all quadratic equations.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
This is a quadratic equation of the form \(ax^2 + bx + c = 0\). The solutions, also known as roots, are the values of x that satisfy the equation.
Step 2: Key Formula or Approach
We can solve this quadratic equation by factoring. We need to find two numbers that multiply to 'c' (-5) and add to 'b' (-4).
Alternatively, the quadratic formula can be used: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Step 3: Detailed Explanation
Using the factoring method for \(x^2 - 4x - 5 = 0\):
We need two numbers that multiply to -5 and add to -4. Let's list the factors of -5:
(1, -5) and (-1, 5).
Let's check their sums:
1 + (-5) = -4.
-1 + 5 = 4.
The correct pair is 1 and -5. So, we can factor the equation as:
\[ (x + 1)(x - 5) = 0 \] For the product of two factors to be zero, at least one of the factors must be zero.
Set each factor to zero to find the solutions:
\[ x + 1 = 0 \implies x = -1 \] \[ x - 5 = 0 \implies x = 5 \] Step 4: Final Answer
The solutions to the equation are x = -1 and x = 5.