Certainly! Let's go through the expression step-by-step.
We are given the quadratic expression:
[tex]\[ -a^2 + 2a + 35 \][/tex]
### Step 1: Identify the Form
This is a quadratic expression in the form [tex]\( ax^2 + bx + c \)[/tex], where:
[tex]\[ a = -1, \quad b = 2, \quad c = 35 \][/tex]
### Step 2: Determine the Axis of Symmetry
The axis of symmetry for a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] can be found using:
[tex]\[ a = \frac{-b}{2a} \][/tex]
Plugging in the values from our expression:
[tex]\[ a = \frac{-2}{2(-1)} = \frac{-2}{-2} = 1 \][/tex]
### Step 3: Identify the Vertex
The vertex of the parabola described by the quadratic expression can be found by substituting the x-coordinate of the axis of symmetry (which is [tex]\( a = 1 \)[/tex]) back into the expression.
Substitute [tex]\( a = 1 \)[/tex] into:
[tex]\[ -a^2 + 2a + 35 \][/tex]
[tex]\[ -((1)^2) + 2(1) + 35 = -1 + 2 + 35 = 36 \][/tex]
So, the vertex [tex]\((a, y)\)[/tex] is at:
[tex]\[ (1, 36) \][/tex]
### Step 4: Analyze the Parabola Direction
Since the coefficient of [tex]\( a^2 \)[/tex] is negative, the parabola opens downwards.
### Step 5: Finding the Roots (If Required)
To find the roots, we set the quadratic expression to zero and solve for [tex]\( a \)[/tex]:
[tex]\[ -a^2 + 2a + 35 = 0 \][/tex]
We can use the quadratic formula, [tex]\( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
Where [tex]\( b = 2 \)[/tex], [tex]\( a = -1 \)[/tex], and [tex]\( c = 35 \)[/tex]:
[tex]\[ a = \frac{-2 \pm \sqrt{(2)^2 - 4(-1)(35)}}{2(-1)} \][/tex]
[tex]\[ a = \frac{-2 \pm \sqrt{4 + 140}}{-2} \][/tex]
[tex]\[ a = \frac{-2 \pm \sqrt{144}}{-2} \][/tex]
[tex]\[ a = \frac{-2 \pm 12}{-2} \][/tex]
Thus, the roots are:
[tex]\[ a = \frac{10}{-2} = -5 \][/tex]
[tex]\[ a = \frac{-14}{-2} = 7 \][/tex]
So, the roots of the equation are:
[tex]\[ a = -5 \][/tex]
[tex]\[ a = 7 \][/tex]
### Conclusion
To summarize, the quadratic expression [tex]\( -a^2 + 2a + 35 \)[/tex]:
- Has a vertex at [tex]\( (1, 36) \)[/tex]
- Opens downwards
- Has roots at [tex]\( a = -5 \)[/tex] and [tex]\( a = 7 \)[/tex]
This fully describes and analyzes the given quadratic expression.
