To determine the locations of the roots of the quadratic equation [tex]\( x^2 - 4x - 5 = 0 \)[/tex], we need to find the values of [tex]\( x \)[/tex] for which [tex]\( y \)[/tex] equals zero. Here is a step-by-step explanation of how to solve this:
### Step 1: Identify coefficients
The given quadratic equation is [tex]\( x^2 - 4x - 5 = 0 \)[/tex]. Here, the coefficients are:
- [tex]\( a = 1 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = -4 \)[/tex] (coefficient of [tex]\( x \)[/tex])
- [tex]\( c = -5 \)[/tex] (constant term)
### Step 2: Use the quadratic formula
The quadratic formula to find the roots of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 3: Calculate the discriminant
The discriminant ([tex]\( \Delta \)[/tex]) is part of the quadratic formula under the square root:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the known values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = (-4)^2 - 4(1)(-5) \][/tex]
[tex]\[ \Delta = 16 + 20 \][/tex]
[tex]\[ \Delta = 36 \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex] using the quadratic formula
With the discriminant calculated, substitute back into the quadratic formula:
[tex]\[ x = \frac{-(-4) \pm \sqrt{36}}{2(1)} \][/tex]
[tex]\[ x = \frac{4 \pm 6}{2} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{4 + 6}{2} = \frac{10}{2} = 5 \][/tex]
[tex]\[ x_2 = \frac{4 - 6}{2} = \frac{-2}{2} = -1 \][/tex]
### Step 5: Determine the root locations
Since the solutions for [tex]\( x \)[/tex] are [tex]\( 5 \)[/tex] and [tex]\( -1 \)[/tex], the roots of the quadratic equation are the points where [tex]\( y = 0 \)[/tex]. Thus, the root locations are:
[tex]\[ (-1, 0) \][/tex] and [tex]\[ (5, 0) \][/tex]
### Conclusion
Comparing these locations with the provided options, we find:
A. [tex]\( (-1, 0) \)[/tex] and [tex]\( (5, 0) \)[/tex]
D. [tex]\( (-2, -3) \)[/tex] and [tex]\( (6, -3) \)[/tex]
B. [tex]\( (2, 4) \)[/tex]
E. [tex]\( (0, 0) \)[/tex]
C. [tex]\( (0, 2) \)[/tex]
Thus, the correct option is A. [tex]\( (-1,0) \)[/tex] and [tex]\( (5,0) \)[/tex].
