Get expert advice and community support for your questions on IDNLearn.com. Get prompt and accurate answers to your questions from our experts who are always ready to help.
Sagot :
To solve the given quadratic equation [tex]\( y = -x^2 + 8x + 2 \)[/tex] and find its vertex and solutions (roots), let's go through the detailed steps.
### 1. Find the Vertex of the Parabola
The vertex form of a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] gives us the peak (or trough) of the parabola. The vertex [tex]\((h, k)\)[/tex] can be calculated using the formula:
[tex]\[ h = -\frac{b}{2a} \][/tex]
where [tex]\(a\)[/tex], and [tex]\(b\)[/tex] are coefficients from the equation [tex]\( ax^2 + bx + c \)[/tex].
For the given equation [tex]\( y = -x^2 + 8x + 2 \)[/tex]:
- [tex]\(a = -1\)[/tex]
- [tex]\(b = 8\)[/tex]
Using the formula:
[tex]\[ h = -\frac{8}{2(-1)} = -\frac{8}{-2} = 4 \][/tex]
To find [tex]\(k\)[/tex], substitute [tex]\(x = 4\)[/tex] back into the equation:
[tex]\[ k = -4^2 + 8 \cdot 4 + 2 = -16 + 32 + 2 = 18 \][/tex]
Therefore, the vertex is:
[tex]\[ (4, 18) \][/tex]
### 2. Find the Roots of the Quadratic Equation
To find the roots (solutions) of the quadratic equation [tex]\( y = -x^2 + 8x + 2 \)[/tex], set [tex]\( y = 0 \)[/tex]:
[tex]\[ -x^2 + 8x + 2 = 0 \][/tex]
This is a standard quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. To find the roots, use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the given equation:
- [tex]\(a = -1\)[/tex]
- [tex]\(b = 8\)[/tex]
- [tex]\(c = 2\)[/tex]
First, compute the discriminant:
[tex]\[ \Delta = b^2 - 4ac = 8^2 - 4(-1)(2) = 64 + 8 = 72 \][/tex]
Then, apply the quadratic formula:
[tex]\[ x = \frac{-8 \pm \sqrt{72}}{2(-1)} = \frac{-8 \pm 6\sqrt{2}}{-2} \][/tex]
This simplifies to:
[tex]\[ x = 4 \pm 3\sqrt{2} \][/tex]
So, the roots are:
[tex]\[ x = 4 - 3\sqrt{2} \quad \text{and} \quad x = 4 + 3\sqrt{2} \][/tex]
### Summary:
- Vertex: [tex]\((4, 18)\)[/tex]
- Solutions: [tex]\( x = 4 - 3\sqrt{2} \)[/tex] and [tex]\( x = 4 + 3\sqrt{2} \)[/tex]
Thus, the solutions and vertex of the quadratic equation [tex]\( y = -x^2 + 8x + 2 \)[/tex] are:
[tex]\[ \left(4 - 3\sqrt{2}, 0\right) \quad \text{and} \quad \left(4 + 3\sqrt{2}, 0\right) \][/tex]
with the vertex:
[tex]\[ (4, 18) \][/tex]
### 1. Find the Vertex of the Parabola
The vertex form of a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] gives us the peak (or trough) of the parabola. The vertex [tex]\((h, k)\)[/tex] can be calculated using the formula:
[tex]\[ h = -\frac{b}{2a} \][/tex]
where [tex]\(a\)[/tex], and [tex]\(b\)[/tex] are coefficients from the equation [tex]\( ax^2 + bx + c \)[/tex].
For the given equation [tex]\( y = -x^2 + 8x + 2 \)[/tex]:
- [tex]\(a = -1\)[/tex]
- [tex]\(b = 8\)[/tex]
Using the formula:
[tex]\[ h = -\frac{8}{2(-1)} = -\frac{8}{-2} = 4 \][/tex]
To find [tex]\(k\)[/tex], substitute [tex]\(x = 4\)[/tex] back into the equation:
[tex]\[ k = -4^2 + 8 \cdot 4 + 2 = -16 + 32 + 2 = 18 \][/tex]
Therefore, the vertex is:
[tex]\[ (4, 18) \][/tex]
### 2. Find the Roots of the Quadratic Equation
To find the roots (solutions) of the quadratic equation [tex]\( y = -x^2 + 8x + 2 \)[/tex], set [tex]\( y = 0 \)[/tex]:
[tex]\[ -x^2 + 8x + 2 = 0 \][/tex]
This is a standard quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. To find the roots, use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the given equation:
- [tex]\(a = -1\)[/tex]
- [tex]\(b = 8\)[/tex]
- [tex]\(c = 2\)[/tex]
First, compute the discriminant:
[tex]\[ \Delta = b^2 - 4ac = 8^2 - 4(-1)(2) = 64 + 8 = 72 \][/tex]
Then, apply the quadratic formula:
[tex]\[ x = \frac{-8 \pm \sqrt{72}}{2(-1)} = \frac{-8 \pm 6\sqrt{2}}{-2} \][/tex]
This simplifies to:
[tex]\[ x = 4 \pm 3\sqrt{2} \][/tex]
So, the roots are:
[tex]\[ x = 4 - 3\sqrt{2} \quad \text{and} \quad x = 4 + 3\sqrt{2} \][/tex]
### Summary:
- Vertex: [tex]\((4, 18)\)[/tex]
- Solutions: [tex]\( x = 4 - 3\sqrt{2} \)[/tex] and [tex]\( x = 4 + 3\sqrt{2} \)[/tex]
Thus, the solutions and vertex of the quadratic equation [tex]\( y = -x^2 + 8x + 2 \)[/tex] are:
[tex]\[ \left(4 - 3\sqrt{2}, 0\right) \quad \text{and} \quad \left(4 + 3\sqrt{2}, 0\right) \][/tex]
with the vertex:
[tex]\[ (4, 18) \][/tex]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.