Let's solve both problems step by step.
Problem 2: Complete the square for \( y = -x^2 + 8x + 7 \)
Step 1: Factor out the negative sign from the quadratic terms
\[ y = - (x^2 - 8x) + 7 \]
Step 2: Complete the square inside the parentheses
To complete the square, we need to take half of the coefficient of \( x \) (which is -8), square it and add and subtract that value inside the parentheses.
Half of -8 is -4. Squaring -4 gives us 16.
So, we rewrite the expression as follows:
\[ y = - (x^2 - 8x + 16 - 16) + 7 \]
This simplifies to:
\[ y = - ( (x - 4)^2 - 16) + 7 \]
Step 3: Distribute the negative sign
Now, distribute the negative sign and simplify:
\[ y = - (x - 4)^2 + 16 + 7 \] \[ y = - (x - 4)^2 + 23 \]
Step 4: Identify the vertex and axis of symmetry
The vertex form of a quadratic function is \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex.
From our equation, we have:
- Vertex \( (h, k) = (4, 23) \)
- Axis of symmetry: \( x = h = 4 \)
Problem 3: Find the zeros of the quadratic equation \( y = 6x^2 + 13x + 5 \)
Step 1: Apply the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our quadratic \( ax^2 + bx + c \) where \( a = 6 \), \( b = 13 \), and \( c = 5 \):
Step 2: Calculate the discriminant \( b^2 - 4ac \)
\[ b^2 = 13^2 = 169 \] \[ 4ac = 4(6)(5) = 120 \] \[ b^2 - 4ac = 169 - 120 = 49 \]
Step 3: Substitute values into the quadratic formula
Since the discriminant is positive, we have two distinct real roots.
\[ x = \frac{-13 \pm \sqrt{49}}{2 \cdot 6} = \frac{-13 \pm 7}{12} \]
Step 4: Calculate the two solutions
-
Using \( + \): \[ x = \frac{-13 + 7}{12} = \frac{-6}{12} = -\frac{1}{2} \]
-
Using \( - \): \[ x = \frac{-13 - 7}{12} = \frac{-20}{12} = -\frac{5}{3} \]
Final Roots: The zeros are \( x = -\frac{1}{2} \) and \( x = -\frac{5}{3} \).
Now for handwriting, you will need to write these steps clearly on paper. You can upload the handwritten work as images labeled as #2 for the vertex and axis of symmetry and #3 for the zeros of the quadratic equation.
If you need further assistance or a summary, feel free to ask!