Question
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.
Desmos Calculator (click here)
2. Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7
. You must use the "complete the square method"
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
(6 points)
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.
Desmos Calculator (click here)
2. Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7
. You must use the "complete the square method"
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
(6 points)
Answers
GPT-4o mini
Sure! Here are the steps for solving the questions you've provided.
### Question 2: Complete the Square
We need to determine the vertex and axis of symmetry of the quadratic function \( y = -x^2 + 8x + 7 \) by completing the square.
1. **Start with the given equation:**
\[
y = -x^2 + 8x + 7
\]
2. **Factor out the coefficient of \( x^2 \) from the \( x \) terms:**
\[
y = -(x^2 - 8x) + 7
\]
3. **Complete the square for the expression in parentheses:**
- Take half of the coefficient of \( x \) (which is \( -8 \)), square it, and add and subtract this value inside the parentheses. Half of \( -8 \) is \( -4 \).
- Squaring \( -4 \) gives \( 16 \).
Add and subtract \( 16 \) inside the parentheses:
\[
y = -(x^2 - 8x + 16 - 16) + 7
\]
\[
y = -[(x-4)^2 - 16] + 7
\]
4. **Simplify the equation:**
\[
y = - (x - 4)^2 + 16 + 7
\]
\[
y = - (x - 4)^2 + 23
\]
5. **Identify the vertex and axis of symmetry:**
- The vertex form is \( y = a(x-h)^2 + k \) where (h,k) is the vertex.
- Here \( h = 4 \) and \( k = 23 \), so the vertex is \( (4, 23) \).
- The axis of symmetry is \( x = h = 4 \).
### Question 3: Find the Zeros
For the quadratic function \( y = 6x^2 + 13x + 5 \), we will use the quadratic formula.
1. **Identify the coefficients:**
- \( a = 6 \)
- \( b = 13 \)
- \( c = 5 \)
2. **Use the quadratic formula:**
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
3. **Calculate the discriminant:**
\[
b^2 - 4ac = 13^2 - 4(6)(5) = 169 - 120 = 49
\]
4. **Apply the quadratic formula:**
\[
x = \frac{-13 \pm \sqrt{49}}{2 \cdot 6}
\]
\[
x = \frac{-13 \pm 7}{12}
\]
5. **Calculate the two solutions:**
- For \( -13 + 7 = -6 \):
\[
x_1 = \frac{-6}{12} = -\frac{1}{2}
\]
- For \( -13 - 7 = -20 \):
\[
x_2 = \frac{-20}{12} = -\frac{5}{3}
\]
6. **The zeros of the quadratic are:**
\[
x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{5}{3}
\]
To complete the response as per your request, please consider that I can't create actual images or handwritten notes. However, you can use the above synthesis of steps to write it out by hand and create images or scan the completed work for submission. If needed, let me know if there’s anything else you want!
### Question 2: Complete the Square
We need to determine the vertex and axis of symmetry of the quadratic function \( y = -x^2 + 8x + 7 \) by completing the square.
1. **Start with the given equation:**
\[
y = -x^2 + 8x + 7
\]
2. **Factor out the coefficient of \( x^2 \) from the \( x \) terms:**
\[
y = -(x^2 - 8x) + 7
\]
3. **Complete the square for the expression in parentheses:**
- Take half of the coefficient of \( x \) (which is \( -8 \)), square it, and add and subtract this value inside the parentheses. Half of \( -8 \) is \( -4 \).
- Squaring \( -4 \) gives \( 16 \).
Add and subtract \( 16 \) inside the parentheses:
\[
y = -(x^2 - 8x + 16 - 16) + 7
\]
\[
y = -[(x-4)^2 - 16] + 7
\]
4. **Simplify the equation:**
\[
y = - (x - 4)^2 + 16 + 7
\]
\[
y = - (x - 4)^2 + 23
\]
5. **Identify the vertex and axis of symmetry:**
- The vertex form is \( y = a(x-h)^2 + k \) where (h,k) is the vertex.
- Here \( h = 4 \) and \( k = 23 \), so the vertex is \( (4, 23) \).
- The axis of symmetry is \( x = h = 4 \).
### Question 3: Find the Zeros
For the quadratic function \( y = 6x^2 + 13x + 5 \), we will use the quadratic formula.
1. **Identify the coefficients:**
- \( a = 6 \)
- \( b = 13 \)
- \( c = 5 \)
2. **Use the quadratic formula:**
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
3. **Calculate the discriminant:**
\[
b^2 - 4ac = 13^2 - 4(6)(5) = 169 - 120 = 49
\]
4. **Apply the quadratic formula:**
\[
x = \frac{-13 \pm \sqrt{49}}{2 \cdot 6}
\]
\[
x = \frac{-13 \pm 7}{12}
\]
5. **Calculate the two solutions:**
- For \( -13 + 7 = -6 \):
\[
x_1 = \frac{-6}{12} = -\frac{1}{2}
\]
- For \( -13 - 7 = -20 \):
\[
x_2 = \frac{-20}{12} = -\frac{5}{3}
\]
6. **The zeros of the quadratic are:**
\[
x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{5}{3}
\]
To complete the response as per your request, please consider that I can't create actual images or handwritten notes. However, you can use the above synthesis of steps to write it out by hand and create images or scan the completed work for submission. If needed, let me know if there’s anything else you want!