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Sagot :
Certainly! Let's analyze the quadratic equation given by [tex]\( y = -x^2 - 8x - 11 \)[/tex].
### 1. Finding the Vertex
The vertex of a quadratic equation in the form [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formulas for [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
1. [tex]\( h = -\frac{b}{2a} \)[/tex]
2. [tex]\( k = y(h) \)[/tex]
Here, [tex]\( a = -1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -11 \)[/tex].
Calculating [tex]\( h \)[/tex]:
[tex]\[ h = -\frac{-8}{2 \cdot -1} \][/tex]
[tex]\[ h = \frac{8}{-2} \][/tex]
[tex]\[ h = -4 \][/tex]
Calculating [tex]\( k \)[/tex] by substituting [tex]\( h \)[/tex] back into the equation:
[tex]\[ k = -(-4)^2 - 8(-4) - 11 \][/tex]
[tex]\[ k = -16 + 32 - 11 \][/tex]
[tex]\[ k = 5 \][/tex]
Thus, the vertex of the parabola is at [tex]\( (-4, 5) \)[/tex].
### 2. Finding the Roots
To find the roots (where [tex]\( y = 0 \)[/tex]), we solve the equation [tex]\( -x^2 - 8x - 11 = 0 \)[/tex]. This can be done using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = -1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -11 \)[/tex].
First, we find the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-8)^2 - 4(-1)(-11) \][/tex]
[tex]\[ \Delta = 64 - 44 \][/tex]
[tex]\[ \Delta = 20 \][/tex]
Since the discriminant is positive, there are two real roots. We can now calculate them:
Root 1:
[tex]\[ x_1 = \frac{-(-8) + \sqrt{20}}{2(-1)} \][/tex]
[tex]\[ x_1 = \frac{8 + \sqrt{20}}{-2} \][/tex]
[tex]\[ x_1 = \frac{8 + 4.472}{-2} \][/tex]
[tex]\[ x_1 = \frac{12.472}{-2} \][/tex]
[tex]\[ x_1 \approx -6.236 \][/tex]
Root 2:
[tex]\[ x_2 = \frac{-(-8) - \sqrt{20}}{2(-1)} \][/tex]
[tex]\[ x_2 = \frac{8 - \sqrt{20}}{-2} \][/tex]
[tex]\[ x_2 = \frac{8 - 4.472}{-2} \][/tex]
[tex]\[ x_2 = \frac{3.528}{-2} \][/tex]
[tex]\[ x_2 \approx -1.764 \][/tex]
Thus, the roots of the quadratic equation are approximately [tex]\( -6.236 \)[/tex] and [tex]\( -1.764 \)[/tex].
### Summary
From our calculations, we have:
- The vertex of the parabola is at [tex]\( (-4, 5) \)[/tex].
- The roots of the parabola are approximately [tex]\( -6.236 \)[/tex] and [tex]\( -1.764 \)[/tex].
These results provide a complete analysis of the given quadratic equation [tex]\( y = -x^2 - 8x - 11 \)[/tex].
### 1. Finding the Vertex
The vertex of a quadratic equation in the form [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formulas for [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
1. [tex]\( h = -\frac{b}{2a} \)[/tex]
2. [tex]\( k = y(h) \)[/tex]
Here, [tex]\( a = -1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -11 \)[/tex].
Calculating [tex]\( h \)[/tex]:
[tex]\[ h = -\frac{-8}{2 \cdot -1} \][/tex]
[tex]\[ h = \frac{8}{-2} \][/tex]
[tex]\[ h = -4 \][/tex]
Calculating [tex]\( k \)[/tex] by substituting [tex]\( h \)[/tex] back into the equation:
[tex]\[ k = -(-4)^2 - 8(-4) - 11 \][/tex]
[tex]\[ k = -16 + 32 - 11 \][/tex]
[tex]\[ k = 5 \][/tex]
Thus, the vertex of the parabola is at [tex]\( (-4, 5) \)[/tex].
### 2. Finding the Roots
To find the roots (where [tex]\( y = 0 \)[/tex]), we solve the equation [tex]\( -x^2 - 8x - 11 = 0 \)[/tex]. This can be done using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = -1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -11 \)[/tex].
First, we find the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-8)^2 - 4(-1)(-11) \][/tex]
[tex]\[ \Delta = 64 - 44 \][/tex]
[tex]\[ \Delta = 20 \][/tex]
Since the discriminant is positive, there are two real roots. We can now calculate them:
Root 1:
[tex]\[ x_1 = \frac{-(-8) + \sqrt{20}}{2(-1)} \][/tex]
[tex]\[ x_1 = \frac{8 + \sqrt{20}}{-2} \][/tex]
[tex]\[ x_1 = \frac{8 + 4.472}{-2} \][/tex]
[tex]\[ x_1 = \frac{12.472}{-2} \][/tex]
[tex]\[ x_1 \approx -6.236 \][/tex]
Root 2:
[tex]\[ x_2 = \frac{-(-8) - \sqrt{20}}{2(-1)} \][/tex]
[tex]\[ x_2 = \frac{8 - \sqrt{20}}{-2} \][/tex]
[tex]\[ x_2 = \frac{8 - 4.472}{-2} \][/tex]
[tex]\[ x_2 = \frac{3.528}{-2} \][/tex]
[tex]\[ x_2 \approx -1.764 \][/tex]
Thus, the roots of the quadratic equation are approximately [tex]\( -6.236 \)[/tex] and [tex]\( -1.764 \)[/tex].
### Summary
From our calculations, we have:
- The vertex of the parabola is at [tex]\( (-4, 5) \)[/tex].
- The roots of the parabola are approximately [tex]\( -6.236 \)[/tex] and [tex]\( -1.764 \)[/tex].
These results provide a complete analysis of the given quadratic equation [tex]\( y = -x^2 - 8x - 11 \)[/tex].
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