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To analyze the quadratic equation [tex]\( y = -x^2 + 6x + 3 \)[/tex], we will find the vertex of the parabola, determine the roots of the equation, and then summarize our results.
### Step 1: Determine the Vertex of the Parabola
A quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] forms a parabola when graphed. The vertex of a parabola can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our quadratic equation [tex]\( y = -x^2 + 6x + 3 \)[/tex]:
- The coefficient [tex]\( a = -1 \)[/tex]
- The coefficient [tex]\( b = 6 \)[/tex]
- The constant [tex]\( c = 3 \)[/tex]
So, the x-coordinate of the vertex is:
[tex]\[ x = -\frac{6}{2(-1)} = \frac{6}{2} = 3 \][/tex]
Next, we find the y-coordinate of the vertex by substituting [tex]\( x = 3 \)[/tex] back into the equation:
[tex]\[ y = - (3)^2 + 6(3) + 3 \][/tex]
[tex]\[ y = -9 + 18 + 3 \][/tex]
[tex]\[ y = 12 \][/tex]
Thus, the vertex of the parabola is:
[tex]\[ (3, 12) \][/tex]
### Step 2: Determining the Roots
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
[tex]\[ \text{Discriminant} = 6^2 - 4(-1)(3) \][/tex]
[tex]\[ \text{Discriminant} = 36 + 12 \][/tex]
[tex]\[ \text{Discriminant} = 48 \][/tex]
The quadratic formula becomes:
[tex]\[ x = \frac{-6 \pm \sqrt{48}}{2(-1)} \][/tex]
Simplify the square root of 48:
[tex]\[ \sqrt{48} = 4\sqrt{3} \][/tex]
So, we have:
[tex]\[ x = \frac{-6 \pm 4\sqrt{3}}{-2} \][/tex]
[tex]\[ x = 3 \mp 2\sqrt{3} \][/tex]
Thus, the roots of the quadratic equation are:
[tex]\[ x_1 = 3 - 2\sqrt{3} \approx -0.4641 \][/tex]
[tex]\[ x_2 = 3 + 2\sqrt{3} \approx 6.4641 \][/tex]
### Summary of Results
- Vertex: [tex]\( (3.0, 12.0) \)[/tex]
- Roots: [tex]\( -0.4641 \)[/tex] and [tex]\( 6.4641 \)[/tex]
These results give a complete analysis of the quadratic equation [tex]\( y = -x^2 + 6x + 3 \)[/tex]. The vertex shows the highest point on the parabola, indicating its maximum value, and the roots are the points where the parabola crosses the x-axis.
### Step 1: Determine the Vertex of the Parabola
A quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] forms a parabola when graphed. The vertex of a parabola can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our quadratic equation [tex]\( y = -x^2 + 6x + 3 \)[/tex]:
- The coefficient [tex]\( a = -1 \)[/tex]
- The coefficient [tex]\( b = 6 \)[/tex]
- The constant [tex]\( c = 3 \)[/tex]
So, the x-coordinate of the vertex is:
[tex]\[ x = -\frac{6}{2(-1)} = \frac{6}{2} = 3 \][/tex]
Next, we find the y-coordinate of the vertex by substituting [tex]\( x = 3 \)[/tex] back into the equation:
[tex]\[ y = - (3)^2 + 6(3) + 3 \][/tex]
[tex]\[ y = -9 + 18 + 3 \][/tex]
[tex]\[ y = 12 \][/tex]
Thus, the vertex of the parabola is:
[tex]\[ (3, 12) \][/tex]
### Step 2: Determining the Roots
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
[tex]\[ \text{Discriminant} = 6^2 - 4(-1)(3) \][/tex]
[tex]\[ \text{Discriminant} = 36 + 12 \][/tex]
[tex]\[ \text{Discriminant} = 48 \][/tex]
The quadratic formula becomes:
[tex]\[ x = \frac{-6 \pm \sqrt{48}}{2(-1)} \][/tex]
Simplify the square root of 48:
[tex]\[ \sqrt{48} = 4\sqrt{3} \][/tex]
So, we have:
[tex]\[ x = \frac{-6 \pm 4\sqrt{3}}{-2} \][/tex]
[tex]\[ x = 3 \mp 2\sqrt{3} \][/tex]
Thus, the roots of the quadratic equation are:
[tex]\[ x_1 = 3 - 2\sqrt{3} \approx -0.4641 \][/tex]
[tex]\[ x_2 = 3 + 2\sqrt{3} \approx 6.4641 \][/tex]
### Summary of Results
- Vertex: [tex]\( (3.0, 12.0) \)[/tex]
- Roots: [tex]\( -0.4641 \)[/tex] and [tex]\( 6.4641 \)[/tex]
These results give a complete analysis of the quadratic equation [tex]\( y = -x^2 + 6x + 3 \)[/tex]. The vertex shows the highest point on the parabola, indicating its maximum value, and the roots are the points where the parabola crosses the x-axis.
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