Question

Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7
.

3. Find the zeros of the quadratic equation y=6x2+13x+5
. You may use any of the following methods: factoring, square roots, complete the square, or quadratic formula.

Please show all steps to solve these two questions. Upload a picture of your work for each and clearly label them as #2 and #3. You may use Desmos to check your work, but you must upload your handwritten work to receive credit.

Answer 1

Sure! Let’s complete the square for the first equation and find the vertex and axis of symmetry. Then we’ll find the zeros of the quadratic equation in the second part.

### Problem 2: Complete the square for \( y = -x^2 + 8x + 7 \)

1. **Start with the function:**
\[
y = -x^2 + 8x + 7
\]

2. **Factor out the coefficient of \( x^2 \) (which is -1):**
\[
y = -1(x^2 - 8x) + 7
\]

3. **Complete the square:**
- Take half of the coefficient of \( x \) (which is -8), square it, and add/subtract inside the parentheses:
- Half of -8 is -4, and squaring it gives us 16.

\[
y = -1(x^2 - 8x + 16 - 16) + 7
\]
\[
y = -1((x - 4)^2 - 16) + 7
\]

4. **Simplify:**
\[
y = -1(x - 4)^2 + 16 + 7
\]
\[
y = -1(x - 4)^2 + 23
\]

5. **Identify the vertex:**
The vertex form of the equation is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex.

- Here, \( h = 4 \) and \( k = 23 \), so the vertex is \( (4, 23) \).

6. **Determine the axis of symmetry:**
The axis of symmetry can be found using \( x = h \).

- So, the axis of symmetry is \( x = 4 \).

### Answer for Problem 2:
- Vertex: \( (4, 23) \)
- Axis of symmetry: \( x = 4 \)

---

### Problem 3: Find the zeros of \( y = 6x^2 + 13x + 5 \)

1. **Use the quadratic formula:**
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 6 \), \( b = 13 \), and \( c = 5 \).

2. **Calculate the discriminant:**
\[
b^2 - 4ac = 13^2 - 4 \cdot 6 \cdot 5
\]
\[
= 169 - 120 = 49
\]

3. **Substituting into the formula:**
\[
x = \frac{-13 \pm \sqrt{49}}{2 \cdot 6}
\]
\[
x = \frac{-13 \pm 7}{12}
\]

4. **Calculate the two possible values for \( x \)**:
- For \( x_1 \):
\[
x_1 = \frac{-13 + 7}{12} = \frac{-6}{12} = -\frac{1}{2}
\]
- For \( x_2 \):
\[
x_2 = \frac{-13 - 7}{12} = \frac{-20}{12} = -\frac{5}{3}
\]

### Answer for Problem 3:
- Zeros: \( x = -\frac{1}{2}, x = -\frac{5}{3} \)

The final step is to write these down on paper, show the calculations clearly, and upload images labeled as #2 and #3, as requested. If you need assistance formatting that or have further questions, feel free to ask!