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1) Solve `x^{2}-4x-5=0` and enter solutions below. please do correctly
Answers
Answered by
https://askanewquestion.com/
now graph it please
Answered by
GPT-4o mini
To solve the quadratic equation \(x^{2}-4x-5=0\), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = -4\), and \(c = -5\).
Now, first, we calculate the discriminant \(b^2 - 4ac\):
\[
b^2 - 4ac = (-4)^2 - 4(1)(-5) = 16 + 20 = 36
\]
Now we plug the values into the quadratic formula:
\[
x = \frac{-(-4) \pm \sqrt{36}}{2(1)} = \frac{4 \pm 6}{2}
\]
This gives us two solutions:
1. \(x = \frac{4 + 6}{2} = \frac{10}{2} = 5\)
2. \(x = \frac{4 - 6}{2} = \frac{-2}{2} = -1\)
Thus, the solutions to the equation \(x^{2}-4x-5=0\) are:
\[
x = 5 \quad \text{and} \quad x = -1
\]
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = -4\), and \(c = -5\).
Now, first, we calculate the discriminant \(b^2 - 4ac\):
\[
b^2 - 4ac = (-4)^2 - 4(1)(-5) = 16 + 20 = 36
\]
Now we plug the values into the quadratic formula:
\[
x = \frac{-(-4) \pm \sqrt{36}}{2(1)} = \frac{4 \pm 6}{2}
\]
This gives us two solutions:
1. \(x = \frac{4 + 6}{2} = \frac{10}{2} = 5\)
2. \(x = \frac{4 - 6}{2} = \frac{-2}{2} = -1\)
Thus, the solutions to the equation \(x^{2}-4x-5=0\) are:
\[
x = 5 \quad \text{and} \quad x = -1
\]
Answered by
GPT-4o mini
I can’t directly create graphical images. However, I can guide you through how to graph the quadratic equation \(y = x^2 - 4x - 5\).
### Steps to Graph the Quadratic Function
1. **Identify Key Features**:
- **Vertex**: The vertex formula for a quadratic \(y = ax^2 + bx + c\) is at \(x = -\frac{b}{2a}\).
- Here, \(a = 1\) and \(b = -4\):
\[
x = -\frac{-4}{2(1)} = \frac{4}{2} = 2
\]
- To find the \(y\)-coordinate of the vertex, substitute \(x = 2\) into the equation:
\[
y = (2)^2 - 4(2) - 5 = 4 - 8 - 5 = -9
\]
- So the vertex is at \((2, -9)\).
2. **Find the Y-Intercept**:
- Set \(x = 0\):
\[
y = 0^2 - 4(0) - 5 = -5
\]
- The y-intercept is \((0, -5)\).
3. **Find the X-Intercepts**:
- We already calculated the x-intercepts when solving the equation, which are \(x = 5\) and \(x = -1\). So the x-intercepts are \((5, 0)\) and \((-1, 0)\).
4. **Plot Points**:
- Plot the vertex \((2, -9)\), y-intercept \((0, -5)\), and x-intercepts \((-1, 0)\) and \((5, 0)\).
5. **Sketch the Parabola**:
- Draw the parabolic curve through the points plotted. Since the leading coefficient \(a = 1\) is positive, the parabola opens upwards.
### Summary of Important Points:
- Vertex: \((2, -9)\)
- Y-Intercept: \((0, -5)\)
- X-Intercepts: \((-1, 0)\) and \((5, 0)\)
You can use graphing software or graphing calculators to draw a more accurate representation by inputting the equation \(y = x^2 - 4x - 5\).
### Steps to Graph the Quadratic Function
1. **Identify Key Features**:
- **Vertex**: The vertex formula for a quadratic \(y = ax^2 + bx + c\) is at \(x = -\frac{b}{2a}\).
- Here, \(a = 1\) and \(b = -4\):
\[
x = -\frac{-4}{2(1)} = \frac{4}{2} = 2
\]
- To find the \(y\)-coordinate of the vertex, substitute \(x = 2\) into the equation:
\[
y = (2)^2 - 4(2) - 5 = 4 - 8 - 5 = -9
\]
- So the vertex is at \((2, -9)\).
2. **Find the Y-Intercept**:
- Set \(x = 0\):
\[
y = 0^2 - 4(0) - 5 = -5
\]
- The y-intercept is \((0, -5)\).
3. **Find the X-Intercepts**:
- We already calculated the x-intercepts when solving the equation, which are \(x = 5\) and \(x = -1\). So the x-intercepts are \((5, 0)\) and \((-1, 0)\).
4. **Plot Points**:
- Plot the vertex \((2, -9)\), y-intercept \((0, -5)\), and x-intercepts \((-1, 0)\) and \((5, 0)\).
5. **Sketch the Parabola**:
- Draw the parabolic curve through the points plotted. Since the leading coefficient \(a = 1\) is positive, the parabola opens upwards.
### Summary of Important Points:
- Vertex: \((2, -9)\)
- Y-Intercept: \((0, -5)\)
- X-Intercepts: \((-1, 0)\) and \((5, 0)\)
You can use graphing software or graphing calculators to draw a more accurate representation by inputting the equation \(y = x^2 - 4x - 5\).