Question
Complete the square to solve the following quadratic equation: y= 4x2−12x−9
𝑦
=
4
𝑥
2
−
12
𝑥
−
9
(1 point)
𝑦
=
4
𝑥
2
−
12
𝑥
−
9
(1 point)
Answers
Answer
x = -0.62132
x = -0.62132
x = 3.62132
x = 3.62132
x = -0.62132, 3.62132
pick
x = -0.62132
x = 3.62132
x = 3.62132
x = -0.62132, 3.62132
pick
Answered by
GPT-4o mini
To complete the square for the quadratic equation \( y = 4x^2 - 12x - 9 \), follow these steps:
1. **Factor out the coefficient of \( x^2 \) from the first two terms:**
\[
y = 4(x^2 - 3x) - 9
\]
2. **Complete the square for the expression inside the parentheses:**
- Take the coefficient of \( x \) (which is -3), halve it to get -1.5, and then square it to obtain \( (-1.5)^2 = 2.25 \).
3. **Add and subtract this square inside the parentheses:**
\[
y = 4(x^2 - 3x + 2.25 - 2.25) - 9
\]
4. **Rearranging gives:**
\[
y = 4((x^2 - 3x + 2.25) - 2.25) - 9
\]
5. **Factor the perfect square trinomial:**
\[
y = 4((x - 1.5)^2 - 2.25) - 9
\]
6. **Distribute the 4:**
\[
y = 4(x - 1.5)^2 - 9 - 9
\]
7. **Combine the constants:**
\[
y = 4(x - 1.5)^2 - 18
\]
The quadratic equation in completed square form is:
\[
y = 4(x - 1.5)^2 - 18
\]
This form allows for easier analysis of the vertex and the graph of the parabola represented by the equation. The vertex is at the point \( (1.5, -18) \).
1. **Factor out the coefficient of \( x^2 \) from the first two terms:**
\[
y = 4(x^2 - 3x) - 9
\]
2. **Complete the square for the expression inside the parentheses:**
- Take the coefficient of \( x \) (which is -3), halve it to get -1.5, and then square it to obtain \( (-1.5)^2 = 2.25 \).
3. **Add and subtract this square inside the parentheses:**
\[
y = 4(x^2 - 3x + 2.25 - 2.25) - 9
\]
4. **Rearranging gives:**
\[
y = 4((x^2 - 3x + 2.25) - 2.25) - 9
\]
5. **Factor the perfect square trinomial:**
\[
y = 4((x - 1.5)^2 - 2.25) - 9
\]
6. **Distribute the 4:**
\[
y = 4(x - 1.5)^2 - 9 - 9
\]
7. **Combine the constants:**
\[
y = 4(x - 1.5)^2 - 18
\]
The quadratic equation in completed square form is:
\[
y = 4(x - 1.5)^2 - 18
\]
This form allows for easier analysis of the vertex and the graph of the parabola represented by the equation. The vertex is at the point \( (1.5, -18) \).
Answered by
GPT-4o mini
It appears you are presenting multiple values for \( x \) and asking for a selection among them. The values you've listed are:
1. \( x = -0.62132 \)
2. \( x = 3.62132 \)
3. \( x = -0.62132, 3.62132 \)
If you need to choose one option, it depends on the context of the problem you are working with.
- If you want a single solution, you can choose either of the individual values \( x = -0.62132 \) or \( x = 3.62132 \).
- If you need both solutions for a quadratic inequality or a similar problem, then you should select \( x = -0.62132, 3.62132 \).
Please provide more context or specify what you are trying to achieve, and I can assist you further!
1. \( x = -0.62132 \)
2. \( x = 3.62132 \)
3. \( x = -0.62132, 3.62132 \)
If you need to choose one option, it depends on the context of the problem you are working with.
- If you want a single solution, you can choose either of the individual values \( x = -0.62132 \) or \( x = 3.62132 \).
- If you need both solutions for a quadratic inequality or a similar problem, then you should select \( x = -0.62132, 3.62132 \).
Please provide more context or specify what you are trying to achieve, and I can assist you further!
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