Question
Find the zeros of y = x2 - 8x - 3 by completing
the square.
0 x=-4 + V19
•x= x= 3+V7
0 x=+3
• x= 4 + V19
the square.
0 x=-4 + V19
•x= x= 3+V7
0 x=+3
• x= 4 + V19
Answers
GPT-4o mini
To find the zeros of the function \( y = x^2 - 8x - 3 \) by completing the square, follow these steps:
1. **Start with the equation:**
\[
y = x^2 - 8x - 3
\]
2. **Move the constant term to the other side:**
\[
y + 3 = x^2 - 8x
\]
3. **Complete the square for the right-hand side:**
Take the coefficient of \( x \), which is \(-8\), divide it by 2 to get \(-4\), and square it to get \( 16 \).
4. **Add and subtract \( 16 \):**
\[
y + 3 = (x^2 - 8x + 16) - 16
\]
This can be rewritten as:
\[
y + 3 = (x - 4)^2 - 16
\]
5. **Rearrange the equation:**
\[
y + 3 + 16 = (x - 4)^2
\]
\[
y + 19 = (x - 4)^2
\]
6. **Set \( y \) to 0 to find the zeros:**
\[
0 + 19 = (x - 4)^2
\]
\[
19 = (x - 4)^2
\]
7. **Take the square root of both sides:**
\[
\sqrt{19} = x - 4 \quad \text{or} \quad -\sqrt{19} = x - 4
\]
8. **Solve for \( x \):**
- For \( \sqrt{19} \):
\[
x = 4 + \sqrt{19}
\]
- For \( -\sqrt{19} \):
\[
x = 4 - \sqrt{19}
\]
So the zeros of the function \( y = x^2 - 8x - 3 \) are:
\[
x = 4 + \sqrt{19} \quad \text{and} \quad x = 4 - \sqrt{19}
\]
Among the options you provided, the correct match is:
- \( x = 4 + \sqrt{19} \)
- \( x = 4 - \sqrt{19} \) (the second root is not explicitly listed in your options).
If needed, you could further approximate \( \sqrt{19} \) which is about \( 4.36 \), giving approximate roots \( x \approx 8.36 \) and \( x \approx -0.36 \).
1. **Start with the equation:**
\[
y = x^2 - 8x - 3
\]
2. **Move the constant term to the other side:**
\[
y + 3 = x^2 - 8x
\]
3. **Complete the square for the right-hand side:**
Take the coefficient of \( x \), which is \(-8\), divide it by 2 to get \(-4\), and square it to get \( 16 \).
4. **Add and subtract \( 16 \):**
\[
y + 3 = (x^2 - 8x + 16) - 16
\]
This can be rewritten as:
\[
y + 3 = (x - 4)^2 - 16
\]
5. **Rearrange the equation:**
\[
y + 3 + 16 = (x - 4)^2
\]
\[
y + 19 = (x - 4)^2
\]
6. **Set \( y \) to 0 to find the zeros:**
\[
0 + 19 = (x - 4)^2
\]
\[
19 = (x - 4)^2
\]
7. **Take the square root of both sides:**
\[
\sqrt{19} = x - 4 \quad \text{or} \quad -\sqrt{19} = x - 4
\]
8. **Solve for \( x \):**
- For \( \sqrt{19} \):
\[
x = 4 + \sqrt{19}
\]
- For \( -\sqrt{19} \):
\[
x = 4 - \sqrt{19}
\]
So the zeros of the function \( y = x^2 - 8x - 3 \) are:
\[
x = 4 + \sqrt{19} \quad \text{and} \quad x = 4 - \sqrt{19}
\]
Among the options you provided, the correct match is:
- \( x = 4 + \sqrt{19} \)
- \( x = 4 - \sqrt{19} \) (the second root is not explicitly listed in your options).
If needed, you could further approximate \( \sqrt{19} \) which is about \( 4.36 \), giving approximate roots \( x \approx 8.36 \) and \( x \approx -0.36 \).