Answer :
To solve for the vertex and the x-intercepts of the quadratic equation \( y = x^2 - 4x - 21 \), we will follow these steps:
### Finding the Vertex
The vertex form of a quadratic equation \( y = ax^2 + bx + c \) can be found using the vertex formula. The x-coordinate of the vertex is given by:
[tex]\[ x = \frac{-b}{2a} \][/tex]
For the given equation \( y = x^2 - 4x - 21 \):
- \( a = 1 \)
- \( b = -4 \)
- \( c = -21 \)
Substitute the values of \( a \) and \( b \):
[tex]\[ x = \frac{-(-4)}{2 \cdot 1} = \frac{4}{2} = 2 \][/tex]
Now, to find the y-coordinate of the vertex, we substitute \( x = 2 \) back into the original equation:
[tex]\[ y = (2)^2 - 4(2) - 21 \][/tex]
[tex]\[ y = 4 - 8 - 21 \][/tex]
[tex]\[ y = -25 \][/tex]
Thus, the vertex is at:
[tex]\[ (2, -25) \][/tex]
### Finding the x-Intercepts
The x-intercepts of the equation are found by setting \( y = 0 \) and solving for \( x \):
[tex]\[ 0 = x^2 - 4x - 21 \][/tex]
This can be solved using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
[tex]\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-21)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{16 + 84}}{2} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{100}}{2} \][/tex]
[tex]\[ x = \frac{4 \pm 10}{2} \][/tex]
This yields two solutions:
[tex]\[ x = \frac{4 + 10}{2} = \frac{14}{2} = 7 \][/tex]
[tex]\[ x = \frac{4 - 10}{2} = \frac{-6}{2} = -3 \][/tex]
Thus, the x-intercepts are:
[tex]\[ (7, 0) \][/tex] and [tex]\[ (-3, 0) \][/tex]
### Conclusion
Based on the calculations, the correct choices from the given options are:
- Vertex: \((2, -25)\)
- x-intercepts: \((-3, 0)\) and \((7, 0)\)
So the correct answers are:
- E. Vertex: \((2, -25)\)
- B. x-intercepts: [tex]\((-3, 0)\)[/tex], [tex]\((7, 0)\)[/tex]
### Finding the Vertex
The vertex form of a quadratic equation \( y = ax^2 + bx + c \) can be found using the vertex formula. The x-coordinate of the vertex is given by:
[tex]\[ x = \frac{-b}{2a} \][/tex]
For the given equation \( y = x^2 - 4x - 21 \):
- \( a = 1 \)
- \( b = -4 \)
- \( c = -21 \)
Substitute the values of \( a \) and \( b \):
[tex]\[ x = \frac{-(-4)}{2 \cdot 1} = \frac{4}{2} = 2 \][/tex]
Now, to find the y-coordinate of the vertex, we substitute \( x = 2 \) back into the original equation:
[tex]\[ y = (2)^2 - 4(2) - 21 \][/tex]
[tex]\[ y = 4 - 8 - 21 \][/tex]
[tex]\[ y = -25 \][/tex]
Thus, the vertex is at:
[tex]\[ (2, -25) \][/tex]
### Finding the x-Intercepts
The x-intercepts of the equation are found by setting \( y = 0 \) and solving for \( x \):
[tex]\[ 0 = x^2 - 4x - 21 \][/tex]
This can be solved using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
[tex]\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-21)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{16 + 84}}{2} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{100}}{2} \][/tex]
[tex]\[ x = \frac{4 \pm 10}{2} \][/tex]
This yields two solutions:
[tex]\[ x = \frac{4 + 10}{2} = \frac{14}{2} = 7 \][/tex]
[tex]\[ x = \frac{4 - 10}{2} = \frac{-6}{2} = -3 \][/tex]
Thus, the x-intercepts are:
[tex]\[ (7, 0) \][/tex] and [tex]\[ (-3, 0) \][/tex]
### Conclusion
Based on the calculations, the correct choices from the given options are:
- Vertex: \((2, -25)\)
- x-intercepts: \((-3, 0)\) and \((7, 0)\)
So the correct answers are:
- E. Vertex: \((2, -25)\)
- B. x-intercepts: [tex]\((-3, 0)\)[/tex], [tex]\((7, 0)\)[/tex]
